CORNELL UNIVERSITY LIBRARY GIFT OF Robert Henry Thixrston MATHEMATICS Cornell University Library QA 51.H66 Tables of the prime numbers, and prime f 3 1924 001 533 805 Cornell University Library The original of tliis book is in tlie Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924001533805 TABLES OF THE PRIME NUMBERS, AND PRIME FACTORS OF THE COMPOSITE NUMBERS, FEOM 1 TO 100,000; WITH THE METHODS OF THEIR CONSTRUCTION, EXAMPLES OF THEIR USE. BY EDWARD HINKLEY, A. M. KATHESIS MUIiTIS ABTIBUS PB0DE8T. BALTIMORE: PRINTED FOR THE AUTHOR. 1853. Entered, according to the Act of Congress, in the year 1853, By EDWARD HINKLEY, In the Clerk 's Office of the District Court of Maryland. STEREOTYPED AT THE BALTIMORE TYPE AND STEREOTYPE FOUNDRY, FIELDING LUCAS, JR., PROPRIETOR. PREFACE. This is the first book, made or published in this country, devoted exclusively to the subjects of fri/me rmmbers and prime factors. Its chief objects are, to explain the methods by which these numbers are ascer- tained ; to furnish correct tables of them to a greater extent and better adapted for general use, than any which are admissible into a treatise upon various subjects of arithmetical science ; and to shew, by rules and examples, their practical' application in the speedy solution of some common but important problems. The peculiar property of prime numbers, upon which their ordinary uses mainly depend, is that, while they theinsehes are incapable of being exactly measured by any in- tegral number greater than a unit, tJiey and their products and powers do constitute all the possible measures, above a unit, of all otiier numbers. To render this property extensively available in the solution of problems, and especially in readily distinguishing frac- tions, ratios and incommensurable roots or surds, which are reducible, from those which are irreducible, and reducing those of them, which are reducible, to their low- est terms; and in finding the least common multiple of two or more given numbers ; it is necessary not only that the prime numbers should be distinguished from the com- posite, but that such of the prime as are measures of the composite, respectively, should be also known. And inasmuch as there is no method of determining directly or speedily whether some of the numbers ending in 1, 3, 7 or 9, are composite or prime, or if composite, what their prime factors are ; it is of great advantage, in the saving of time and labor, to have either all the factors of each of the composite num- bers within convenient limits, ascertained and orderly arranged in tables, or such of them, at least, as may afford the means of readily finding the rest. Early in the year 1850, the author of this book undertook to devise some easy method for the ascertainment of prime numbers. He had not then seen any table of such numbers, nor did he know what any author -had written respecting them. His ignorance on the subject was not owing so mtich to neglect to seek information, as to the circumstance of his not being able to obtain it from the persons or books consulted. Finding that prime numbers could not be discovered, to any great extent, by any direct method, and not assuming numbers to be prime, without demonstrating them to be so, he first devised and used, for the discovery of them, to a limited extent, the method by multiplication, as explained in Part I, Chapter II. By the aid of the prime numbers thus ascertained, he invented and constructed tables hke that on page 17, at the first, however, only for the purpose of discovering prime numbers to a greater extent. In the process of constructing them, it was observed, that by inserting the prime numbers extensively and orderly, in the manner described in Part II, Chapter I, he could express in tables of a similar form, all the prime factors of each of the composite nmnbers within their limits, while the development of new prime numbers would not be interrupted. Thereupon he constructed, first, the table of prime factors of odd numbers, and then that of even numbers. After he had constructed these tables and had concluded to publish them with ex- planations and rules for their use, he obtained through the kindness of his much es- teemed friend, Mr. Chas. Polsom, Librarian of the Boston Athenaeum, the use of several mathematical works which give some information upon the subjects of this 3 iv PREFACE. boo^. One of them, entitled "Maseres'MathmuUics," being a. collection of tracts on different subjects by different authors, was found to contain a reprint of Mr. Thomas Brancker's table of incomposite or prime numbers below 100,000. For an account of its origin and character see page 163. Upon a view of this table and upon reading the remarks made upon it by Dr. John Wallis.the author determined to republish all that portion of it not comprised in his own tables. By incorporating the use of this part of it with that of his own tables, he has, by the rules of Part III, furnished very great facilities for finding all the prime factors of all the composite numbers below 100,000 ; of all the even numbers to 200,000 ; and of about seven-tenths of all the odd numbers between 100,000 and 200,000 ; and of many more numbers both odd and even above this limit. In consequence of this addition to his work, as originally designed and prepared for publication, some portions of it, particulai-ly pages 19 and 20 of Part II, and all of Part III, have been altered, and a few things deemed interesting in regard to au- thors, tables, &c., have been annexed by way of Appendix. No pains have been spared to attain perfect accuracy in all the tables. The au- thor's table of factors of odd numbers was verified by actual multiplications of the factors of every composite number within its limits ; Brancker's being found to con- tain errors, was thoroughly revised and corrected ; and lastly, the proof sheets of both were compared with Vega's table of factors, so far as they have any thing in common with it. Their accuracy being secured in stereotype plates will be pre- served in all the editions which may be struck from them. The author is aware that prime numbers have uses which he has not described. Their properties and relations constitute no inconsiderable part of the whole theory of numbers. This is a theme which has engaged the attention of some of the most distinguished mathematicians of Europe ; and which, under their powers of ar.alysis and invention, has been prolific of theorems and formuloe, which may be found in their erudite volumes ; but which are not embraced within the scope or design of this little work. For mathematicians the tables will serve all the purposes to which they are ade- quate, without regard to the rules or examples herein set forth. These are designed for the instruction, not of the learned, but of the ignorant ; those who, for want of books or teachers, are unacquainted, either wholly or in part, with those uses of the tables which the author has explained. In view of this object care has been taken in all the explanations, definitions and rules introduced, to use language at once simple, unambiguous and exact. The processes are inductive as well as demonstrative, in order that the subjects treated of may be interesting and intelligible to the greatest possible number of persons ; even to those who may not be familiar with Algebra or proficient in all the rules of Arithmetic. It is thought that young students of either of these elemental parts of mathematics will find in this book explanations of measures, multiples, ratios, and proportions, simple and compound, and of the methods of stating, in the form of an equation, the terms of questions to be solved by the rules of Proportion, more sim- ple, complete and practical, than those contained in ordinary treatises upon the sci- ence of nuralaers. The author is not aware that any table of factors has been published in the United States except that appended to the Arithmetic of Horace Mann, LL. D. and Pliny E. Chase, A.M., which contains all the factors of the composite numbers odd and even from 1 to 1,000 ; and some of those of the odd numbers from that limit to 12,- 700, as well as all the prime numbers to this extent. Of the existence of this table he had no knowledge until June, 1851, when a great portion of his own tables had been made. It was, indeed, the first table of the kind which he ever saw. Since that time, however, he has seen the tables of Barlow and Vega as well as that of PREFACE. V ^rancker ; and he has read the brief descriptions given by Legendre, in a supplement "to his Essay on the Theory of Numbers, of the very extensive tables of Chernao and of Burckhardt ; some account of which may be seen in the Appendix. The labor of constructing, verifying and revising all the tables now. offered to the public, has not been trifling. It would not have been undertaken by the author, who is advanced in years, feeble in health and almost daily engaged in professional duties, but from a strong sense of their utility and a more than brdinary love of mathematical studies. He trusts that his labor has not been expended in vain. He •confidently believes that this book will fill a space not, to any great extent, pre- occupied in this country ; that it will prove a labor-saving instrument acceptable and available, not only to mathematicians, professors and teachers, but to students of arithmetic in the colleges, academies and common schools, and those who may be .generally or frequently engaged in arithmetical calculations, throughout our broad domains; and be esteemed some small tribute to the science of numbers. EDWARD HINKLEY. Baltimore, Nov. 1, 1852. TABLE OF CONTENTS. PAGE. Introduction 7 PART I. Of the Methods devised for the Discovery of Prime Numbers 12 Chap. I. — To determine the class or classes of numbers among which the prime numbers are to be sought, 12 Chap. II. — To find the composite numbers by multiplication,- 13 Chap. III. — To find the composite numbers by ascertaining their divisibility by some prime number, without actually dividing them by such number, 15 The Form of a Table for discovering prime numbers 17 Remarks upon the manner of constructing this Table 18 Chap. IV. — Remarks upon the annexed Table of prime numbers, 19 PART II. Of the Methods devised for the Discovert of the Prime Factors of Composite Numbers, 21 •Chap. I. — Of the construction of the table of the prime factors of the odd num- bers from 1 to 20,000, 21 Ohap. II. — Of the methods used for constructing the table of the prime factors of the even numbers from 1 to 12,500, 22 PART III. Op the tiSE OP THE Tables in finding the Prime Factors op Numbers within and beyond their limits 25 Chap. I.— Of the means of finding the prime factors of numbers within the limits of the tables, that is, from 1 to 100,000, . . .'' 25 Chap. II. — Of the methods of finding the prime factors of numbers above the limits of the tables 31 PART IV. Examples of the use of Prime Numbers and of Prime Factors in the Solution of Problems 35 Chap. I. — Of multiplication and division 35 Chap. II. — Of the measures of numbers, 43 Chap. III. — Of the multiples of numbers, 51 Chap. IV. — Of ratios and fractions, 57 Chap. V. — Of roots rational and irrational or surd 67 Table of the prime numbers from 1 to 20,000 73 Table of the prime factors of the composite odd numbers to the same extent, . . 95 Table of the prime factors of the even numbers from 1 to 12,500, 137 Description of Brancker's table, 163 Part of Brancker's table, 165—205 Tables of Powers of Prime Numbers 206 — 207 Some Relations of Numbers, 208 Appendix, 209 6 INTRODUCTION. EXPLANATION OF TERMS, SIGNS, &c. Prime numbers could not be discovered without a knowledge of the relations they have to composite numbers. It is important there- fore to obtain a clear perception of their distinct properties. They are usually defined by the process of multiplication or of division ; but not necessarily so, as these rules are only short methods for ex- pressing the results, respectively, of additions and subtractions of equal numbers. Composite numbers are definable either with reference to the man- ner by which they may be composed, or to that by which they may be decomposed, analysed, or tested. Their composition may be effected by addition or multiplication, and their decomposition by subtraction or division. For the better understanding of the manner of indicating and discovering composite numbers by means of their factors, and for the purpose of explaining these from their origin, de- finitions in different forms wiU be given. Jl composite number is a number that may be exactly made up of equal •whole numbers greater than a unit. Jl prime number is a number that cannot be exactly made up of equal whole numbers greater than a unit. The definition of a prime number, though simply negative of the essential property of a composite number, conveys this meaning — that the sum of any two or more equal whole numbers, severally greater than a unit, will always either exceed or fall short of the exact magnitude or quantity of any prime number. A comparison of a few only of the small numbers of the natural series of numbers, 1, 2, 3, 4, 5, 6, Stc. will clearly illustrate the meaning of the foregoing definitions, and enable the reader to under- stand how composite numbers may be expressed or indicated as pro- ducts hy means oi factors. The numbers 1, 2 and 3, are evidently prime numbers. The number, 4, is the least of all composite numbers. It may be made up or composed of two and two or twice two ; and be expressed either as a sum, thus, 2-(-2 ; or as a product, thus, 2-3. The number 5 is prime. 8 INTRODUCTION. The number 6 is composite. It is capable of being made up of equal numbers in two different ways, in reference to each of which it may be expressed either as a sum thus, 2-|-2-f-2 ; and 3-|-3 ; or as a product, thus, 2'3 and 3'2. Each of the equal parts of a composite number is called its measure, or one of its measures. The equal parts or measures of a composite number, expressed with the sign of multiplication between them, are called factors, and their product is always the same thing as the composite number itself. The factors, when their origin is observed, are found to have these different significations, viz. one of them expresses the magnitude of the measure, and the other the number of times such measure is con- tained in the composite number; or, in other words, the number of its measures of that magnitude. The factors of the number 4 are equal, because its measure and the number of its measures are equal. But the factors of the number 6 are unequal, because it has mea- sures of different magnitudes. When one of them is viewed as a measure the other expresses the number of the measures ; and vice versa. This difference of meaning in the factors is but obscurely seen, or wholly lost sight of, in common multiplication, especially as the factors, when different, are usually expressed in one only of their two orders of position, 2*3 or 3'2. Their signification, however, is the same in whichever order they may stand ; and as their products, in the two orders, are equal, their expression once, in either order, is sufficient. The factors of 4 and 6 are limited to two, and each of them is a prime number. But it is important to understand well in what man- ner the factors of a composite number are to be expressed when they contain more than two measures either of the same magnitude, or of different magnitudes. Passing the number 7 as a prime number, let us for a moment ex- amine the number 8. This is found to be a composite number having two measures of different magnitudes, viz. 2 and 4. It may be expressed as a sum under two forms, thus, 2-|-2-|-2-}-2 and 4+4 ; and as a product under three forms thus, 2-4, 4-2 and 2-2-2. This last form virtually includes each of the other two, and may be used as the only necessary form to express all the measures of this number. For we have seen that the number, 4, was expressed by two equal factors, 2-2. The product of these factors, therefore, is always to be viewed as their value; and this may be at any time substituted for them. INTRODUCTION. 9 If a third factor be prefixed or annexed to two, the three ought to express the same value as the product of the two multiplied by the third would do. Hence 2-2-2 = 2-4 = 4-2 = 8. The three equal factors of 8 do indicate its two different measures, when it is considered that any two of them is equivalent to its larger measure. If now to the two different factors of the number 6, we annex a third one which is a prime number and at the same time different in magnitude from either of them, thus, 2"3'.5, we may understand that their equivalent composite number is their product, that is, the pro- duct of any two of them multiplied by the other. For 6*o = 30, 10-3 = 30 and 15-2=30. From this example of three unequal prime factors, as well as that of three equal ones, we learn that a composite factor may be substi- tuted for its equivalent prime factors, and vice versa, such factors may be substituted for their equivalent composite factor. Wherefore, if any or all of the factors of any number are composite, they may be exchanged for their equivalent prime factors. Hence, every composite number may be expressed or indicated by its prime factors only. All the different pripie factors of any number indicate aU its differ- ent measures that are prirne numbers, while all the different products that can be formed by different combinations of those factors, consti- tute all its different measures that are composite numbers. The number of the prime factors of a composite number depends upon its magnitude and that of the factors. When the factors of any number exceed two, they may be reduced to that number ; and when so reduced, one of the two will be a com- posite niynber and the other a prime number^ if the whole nujnber of prime factors does not exceed three; but if it does exceed three, both factors may he composite, or one composite and the other prime, ac- cording as there shall be made two products or only one from the prime factors. For instance, the factors 2-3'5'7 = 6'35, or 30-7. When all the factors of a number are of different magnitudes, they must all be distinctly expressed, and in such case their composite number is denominated simply a, product. But when all the factors of a number are of like or equal magni- iudes, it is usual, for the sake of brevity, to express but one of them, and to indicate their number by an index or exponent thus : 2*^ for 2-2, and 2^ for 2'2"2, and so for any other number. The composite number whose factors are equal, is a product as well as that whose factors are unequal ; but it is also called ^ power of one of its equal factors, while each of these is called its root, or first power. 9 10 INTRODUCTION. The index, in expressing the number of equal factors, indicates the degree or height of the powers ; as 2" is called the second, 2' the third power of 2, &c. The index of the root oi first power, though not ex- pressed, is always understood to be 1. In the same manner, when some only of the factors are like, the number of like factors is expressed by an index, as 2''*3 for 2'2"3=zl2, 3»-7 for 3-3-7 = 63 and 2.^-T for 3-3-7-7 = 441, &c. The manner of indicating the composition of a composite number from its equal parts or measures having been shewn by means of the signs of addition and of multiplication, it is prefer to shew, briefly, how its decomposition, or separation into such parts or measures, may be indicated by the signs of subtraction and of division. If from any composite number, one of its measures or the quantity thereof, be successively taken away as many times as it is therein contained, it is evident that the whole of it wiU be exactly used up, and that there will be no remainder. Such operation and its effect may be expressed by the sign of sub- traction, thus : 4 — 2— 2 = 0, or otherwise 4— (2+2)i=:0. And 6 — 2 — 2 — 2 =z 0, or 6 — (2+2-|-2)= 0. And 6 — 3 — 3 = 0, or 6— (3+3)=0. Or it may be more briefly expressed by the sign of division, in the form of a fraction, thus : -^^=.1, -^z^l, and ^^^ = 1. The division may be actually made by dividing first by one of the factors, and the quotient by another, and so on until every one of the factors shall have been used as a divisor ; thus, in regard to the prime factors of 210 which are 2-3"5-7. It is immaterial in what order the divisors are used. , 2)210 '^^^^ example shews in what manner the prime factors 3 1105 might have been developed or discovered by the pro- r:\ OK cess of division, provided the divisibility of the several -. - dividends were known or could be discovered. Aj- It shews, moreover, that the first divisor and the first quotient taken as factors are equal to the first and se- cond divisors and the second quotient as factors : that is, in this ex- ample, 2-105 = 2-3-35=z: 210; and that at any stage of the process in any similar example, the divisors used and the last of their quotients are factors equivalent to the given number. New definitions may now be framed of a compound character with reference only to multiplication and division, which may serve as rules for testing numbers for the purpose of ascertaining whether they are composite or prime. INTRODUCTION. 11 A composite number is a number that may be produced by multiply- ing together two or more other numbers, which are called its factors, by each of which it inay be divided without a remainder . A prime number is a number that cannot be so produced. It is not divisible by any number other than itself or a unit, without a remainder. Instead of the definition of a composite number just given, the rules for determining a composite number may be separately stated in a more simple form, thus : 1. Every number is composite that may be produced by multiplying together any other numbers. 2. Every number is composite that may be divided by any other number, greater than a unit, without a remainder. Every number that is not composite is prime. By a systematic application of these rules, the tables in this book were constructed. The particular methods devised for their construction and their use in discovering the prime factors of numbers beyond their limits, as well as in the solution of certain arithmetical problems, will now be explained, under four diiFerent heads or parts : viz. I. Of the methods devised for the discovery of prime numbers. II. Of the methods devised for the discovery of the prime factors of composite numbers. III. Of the use of the tables of prime numbers and of prime fac- tors in the discovery of the prime factors of numbers beyond the limits of the tables. IV. Examples of the use of prime numbers and of prime factors in the solution of problems. PART I. OF THE METHODS DEVISED FOR THE DISCOVERY OF PRIME NUMBERS. CHAPTER I. TO DETERMINE THE CLASS OR CLASSES OF NUMBERS AMONG WHICH THE PRIME NUMBERS ARE TO BE SOUGHT. As there is no direct method of finding prime numbers, it is neces- sary first to ascertain all the composite numbers within any prescrib- ed limits, and thereby the prime numbers within those limits will be discovered, every number that is not composite being prime. From the definition of a composite number, it is evident that all the even numbers, 2, 4, 6, 8, 10, 12, &c., except the number 2 only, are composite. They may be successively produced, to any extent, by multiplying each term of the natural series by 2, as a constant factor. Moreover every even number is divisible by 2 without a remainder. The number 2 therefore is the only even prime number. It is easy also to see, that of the series of odd numbers, 1, 3, 5, 7, 9, 11, 13, 15, &c., all those ending in 5, except 5 itself, are composite numbers. They may be successively produced by multiplying each term of the series of odd numbers by 5, as a constant factor. And every number ending in 5 is divisible by 5 without a re- mainder. The number 5 therefore is the only prime number among all the odd numbers that end in 5. The field for discriminating composite from prime numbers, is thus reduced to the numbers that end in 1, 3, 7, and 9, which comprise only four-tenths or two-fifths of the numbers in any portion of the natural series beginning with 1 and ending with an even number. As the digits in the units' places of a series, composed exclusively of numbers of those terminations, are repetends, each of which con- sists of four terms, and the digits in the tens' places are peculiar repetends, each of which consists oi forty terms, it is easy to express 12 DISCOVERY OP i-RIME NUMBERS. 13 all such numbers in a progressive order, thus, 1, 3, 7, 9 ; 11, 13, 17, 19 ; 31, 23, 27, 29 ; &c. Two methods of ascertaining the composite numbers of this series will now be described, one of which may be distinguished as the method by multiplication, the other that by division, or rather by de- tecting and marking the divisibility of the numbers without actually dividing them. CHAPTER II. TO FIND THE COMPOSITE NUMBERS BY MULTIPLICATION. Considering every composite odd number as a product, it is certain that aU its factors must be also odd numbers. For if any one factor of any number be an even number, the product of all the factors will be an even number. It is evident, moreover, that if a composite number terminates in any one of the digits 1, 3, 7, 9, no one of its factors can be the num- ber 5, or any number ending in 5. Hence it is known that aU of the factors of every composite number ending in either one of those digits, must themselves end in some of the same digits, and be comprised in the same series. If therefore every term of a series, composed exclusively of numbers of those terminations, were multiplied successively by every other term thereof, all their possible products, and consequently all the composite numbers in'the series would be produced. In order however to obtain all the composite numbers within the limits of a given portion of such a series, the number of terms neces- sary to be used as factors wiU be only about one-third of the numbers in that portion. It is proposed for example, to find, by multiplication, all the com- posite numbers ending in any one of the digits, 1, 3, 7, 9, from 1 to 100. Omitting the number 1 as not a material factor of a composite num- ber, we shall have only three numbers, viz. 3, 7, 9, of the first repe- tend, but four in each of the other nine repetends, viz. 11, 13, 17, 19, in the second; 21, 23, 27, 29, in the third, as so on to the last num- ber, 99. To determine to what e:^tent and in what manner the terms of this series must be used to solve the problem proposed, it must be con- sidered that the largest product required cannot exceed the number 14 DISCOVERY OP PRIME NUMBERS. 99, the greatest extreme of the given series ; and that, as the least extreme is the number 3, it is not necessary that the greatest factor to be used should exceed one-third of 99, that is 33. To perform the work, the factors are arranged in a small table in the form of a square or parallelogram, the longer set being placed in a perpendicular column on the left margin, and the shorter one in a horizontal line at the top of the table ; the former of which' may be called the multiplicands and the latter the multipliers. In multiplying by the second or any succeeding multiplier, it is unnecessary to express any product less than its second power or square, because a product equal to every such one will have been obtained by the product of the same number, among the multipli- cands, by some preceding or smaller multiplier. For instance, the product of 7 times 3 is the same as that of 3 times T. By observing this rule some duplicate products may be avoided. The work and its results are exhibited in the following table. 3 7 9 3 9 7 21 49 9 27 63 81 11 33 77 99 13 39 91 17 51 19 57 21 63 23 69 27 81 29 87 31 93 33 99 It wiU be observed that there are three duplicate products, one under the multiplier 7, and two under the multiplier 9 ; these last being the only products of this multiplier. The cause of these results is apparent. The number 21 among the multiplicands is a composite number equal to 3-7 ; and when multi- plied by 3, becomes equal to 3'3-7 = 9'7 or 7-9. So the numbers 9 and 27, being respectively the second and third powers of 3, when multiplied respectively by 3, give the products 27 and 81, its third and fourth powers, the latter of which is the same as the product of 9 by 9, in the last column. For a similar reason the duplicate product 99 in the same column was developed. DISCOVERY OF PRIME NUMBERS. 15 It is perceived, by an examination of the results, that the number 9 was an unnecessary factor, as a multiplier, as the product by it had been previously obtained. The duplicate products, however, do not render this method in any degree uncertain. In arranging the products obtained as the com- posite numbers sought, the same number will not be twice set down. The different composite numbers developed in the foregoing table, need not be here arranged, as they may be all seen in the first column of the table of prime factors of odd numbers annexed ; the blank spaces of which -shew also all the prime odd numbers from 1 to 100. [It may be here remarked, that this simple table not only developes all the prime numbers in the first century of numbers (except 2 and 5) but it also furnishes the means of easily distinguishing the prime from the composite numbers. Tor the table shews that every com- posite number ending in 1, 3, 7, or 9 in the century, except three, viz. 49, 77 and 91, is divisible by 3. And as no number is divisible by 3 unless the sum of its digits is so divisible, it is easy, without other means, to determine every prime number in th": first century.] By the same method composite and prime numbers might now be ascertained to a greater extent, with the advantage of using as multi- pliers, prime numbers only, and thereby avoiding some duplicate products. But, inasmuch as by the aid of the prime nunjbers already found, the method by division may be employed with less labor, and its results be more conveniently exhibited in tables, that by multi- plication is here dropped. CHAPTER III. TO FIND THE COMPOSITE NUMBERS BY ASCERTAINING THEIR DIVISI- BILITY BY SOME PRIME NUMBER, WITHOUT ACTUALLY DIVIDING THEM BY SUCH NUMBER. Considering, as before, that all the composite nutabers sought are products, and regarding all their factors as prime numbers, it is evi- dent that every one of them must be divisible by some prime number. As soon therefore as any number is found to be so divisible, it is known to be a composite number, whether its other factors be known or not. Hence, in order to determine whether or not a number is composite or prime, it is only necessary to ascertain with certainty whether it is or is not divisible by any, prime number. The method IQ DISCOVERY OF PRIME NUMBERS. of doing this, in respect to every number in a given series, will now be explained. It is proposed to find all the composite odd numbers from 1 to 999. A blank table is first prepared, with lines and spaces convenient for an orderly arrangement of all the odd numbers within the limits pre- scribed, inclusive of those ending in 5 ; which are included not for the purpose of determining their character, which is already known, but that the composite numbers divisible by each prime number suc- cessively, may stand, in regard to space, at equal-distances from each other in the table, and have, moreover, a common numerical differ- ence in a series. This greatly facilitates the work to be perfonBftdi The order of arrangement is such, that aU the numbers in each successive hundred or century, are assigned to one perpendicular column of blank spaces, at the head of which are inserted proper numbers to indicate the particular century of the column. There being in the problem now to be solved the range of ten cen- turies, ten columns of spaces are marked out. On the left margin of the table are written in a progressive order, from top to bottom, aU the odd numbers consisting of the units and tens of a century, which being written in horizontal spaces that run across the table, are in- tended to serve for every number in each century in the table. By this arrangement the space assigned to each number to be tested is situate at an angle formed by the perpendicular and the horizontal lines. The table being thus prepared, the composite numbers are ascer- tained from their divisibility by some one or more of the odd prime numbers already ascertained, in the following mjinner. The number 3, and every successive prime number, is written in the space of its second power or square and in that of every higher number in the table divisible by such prime number, until a prime number is reached whose square exceeds the largest number in the table. The work is then complete. Every number, in whose space any one or more of the prime num- bers shall have been written, is a composite number; and every number whose space is left blank is a prime number. See the table and the work on the next page. TABLE FOR DISCOVERING PRIME NUMBERS 17 From to 100 100 200 1 200 300 300- 400 <400 500 500 600 600 700 700 800 800 900 900 1000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3 7 5 3 11 7 3 5 3 13'"" 3-5 11 3 5 3.13 3 11 7-5-23 3 17 3-7 5 3 3 5-11 3-7" '" 13 3-5' ' ' " 3-23 7 5 3- 11- 19 17 3 5 7-13 3 19 3-5 ' 7 .... 3-5-7 3 3 3 7 5 3 7 3-19 5 11 3 3 23 5-11-13 3 7 3 5 3 5 7 3 13 3-5' ' ' 3'-5-7' ii"'" 3 17 5-13 3 7 3 5 19 3-7-13 s'.ii'.ii' 11 3-5 7 3 13 5 3 3-5 3 5 " 3 ■ •• ■ 3 5 5 3 7 11 3 5 3 5-17 7 31M3 3-5-7 17 23 3 13 5 3 7-11 3 5 3 7 3-5 3-7-11 5 3 3 17 3-5-7' 11 3 7-17 5 3 7-19 3 5.11-17 3 23 3-5-7 13 3 3 5 3 11 7 3-5 .3-5' ' ' 19 3-7 5 3 3 3-5 " 7""' 3 5 3 3-7' 5 3 11 5 3-7 3 5-7 13 3 3-13-19 29 3 5-13 7-11 3 23 3.5 5 3 7 3 5 3-ii-23 3 19 7 3-5 13"" 3.1M7 5 3-7 3 5-23 11 3-7 5 3 3-13 5 3-7-17 19 3-11 5 11 3 5-7-13 s'-n" 3 5 3 7 11 3-5 7" 3 3-5-19 3-7 5 3-17 11 13 3 5-7 5 3-11-29 7 31 3 5 3-i7-i9 7 3-5-13 3 •13- 17 5-7-19 23 3 11 7 3-5-17 13 3 S-6-11 5 3 3-7-i3 5-11 3 3 3-5' 7 13 3 5-7'" 3 3 7 3-5 " 13 7 3 11 5-19 3 3-5 5 3-7 19 11 3 5 3 7 11 3-5-13 7 3 5 3 13 3'7-ii' 5 17 3 3 11 3 3 5'" 3 7 3 5 3 13 3-7 5 3 5 11 3-7 3-5 7 17 3 5-7-11 3 17 3 5 3- 7! 19 3-5 5 3-7 23 3 5 3 [ 3 17"" 3.5.11 7 19 3 5-7-i7 3 3 7 13 3-5 17 7 3-11 19 5 3-13-23 39 3-5-13 5 3-11 13 3 18 DISCOVERY OF PRIME NUMBERS. FURTHER REMARKS ON THE MANNER OF CONSTRUCTING THE PRE- CEDING TABLE. To determine the proper spaces in which to insert the prime num- bers greater than the number 7, instead of counting the spaces from one divisible number to another, it is better to add continually to its square a sum equal to double the prime number itself; which is the common numerical difference between its composite numbers in the table. The common difference for the number 11 is 22, that for 13 is 26; so that the next number divisible by 11, after its square 121, is found to be 143; and the next divisible by 13, after its square 169, is 195. This method renders the process simple and less liable to error than that of counting spaces equal to the number of units in each prime number. A little reflection will suffice to satisfy the reader, that when the object of the work is to find the composite numbers only, and not at the same time all their prime factors also, it is unnecessary to set down any prime number in the space of any number that is less than the square of such prime number. It must be considered that every composite number has, at the least, two prime factors ; and that its divisibility may be ascertained by using either one of them as a factor. The rule in question insures the insertion of one, at the least, of the factors. If the composite number is a square, its factors are equal, each of them being its square root. Such root, according to the rule, will always be insert- ed. If the composite number is not a square, its prime factors will be unequal, and the smaller one will, under the rule, always be in- serted ; which is sufficient to indicate the divisibility of the number in question. The number 11, for instance, was first inserted in the space of its square 121. Upon inspection of the table, it will be found that each one of the smaller numbers of which it is a factor, viz. 33, 55, 77 and 99, had been detected as a composite number by its respective co- factors 3, 5 and 7. For a similar reason, when the object of the work is limited as above mentioned, it is unnecessary to insert in the table any prime number whose square is greater than the highest number in the table. Each of its co-factors less than itself, having been already inserted, sufficiently indicates every composite number that may have some factor greater than the highest prime number directed to be inserted. The largest prime factor which it was found necessary to insert in the foregoing table is 31. The square of 37, the next larger one, being 1369, which exceeds the highest number in the table. DISCOVERY OF PRIME NUMBERS. 19 CHAPTER IV. OF THE DIFFERENT MODES IN WHICH THE PRIME NUMBERS ARE EXPRESSED IN DIFFERENT TABLES OP THIS BOOK, ETC. The tables in this book contain all the prime numbers from 1 to 100,000 of the series of the natural numbers. Those from 1 to 20,000 (except 2 and 5) are comprised in the author's table of Prime Numbers, and also in his table of the Prime Factors of Odd Numbebs. The rest are comprised in Brancker's Table. In the author's table of Factors, and in Brancker's table as publish- ed in this book, the prime numbers are distinguished by blank spaces with dotted lines. In all these tables the prime numbers have the same relative situa- tion in regard to the composite numbers of the like terminations and in regard to each other; except that in the author's table of prime numbers, as well as in Brancker's, no spaces are allotted for the numbers ending in 5. Although all the prime numbers below 20,000 might be contained, in a condensed form, on four or five pages only, while in the author's table they are spread over twenty pages ; yet this table, in its pecu- liar form, affords various advantages, which could not be wisely dis- pensed with, merely for the sake of economy. In this table the prime numbers are all positively expressed in a distinct and uniform order, so that each page comprises the range of one thousand, and each column of one hundred numbers of the natu- ral series ; and any particular number can be found with great ease and despatch, while their relative situations in regard to each other and to the proximate composite numbers are very clearly seen and understood. The number of prime numbers in each one of aU the centuries, exclusive of the numbers 2 and 5 in the first century, is set down at the foot of its proper column throughout this as well as Brancker's table. This enables us to see their comparative numbers in the dif- ferent centuries as well as to ascertain the sum of all of them in both tables. It appears that the greatest number, 24, or (inclusive of 2 and 5) 26, is in the first century; and that the least number 3, is found in_ two centuries, viz ; the six hundred and ninety-sixth, and the seven hundred and ninety-eighth century. That after the second century no one has a greater number of prime numbers than 17; and that all 20 DISCOVERY OF PRIME NUMBERS. the different numbers in all the centuries, exclusive of the first two, are comprised in the series of numbers from 3 to 17 inclusive. There is a comparatively large number of prime numbers in the first four or five centuries, which in the tables constructed for the discovery of prime numbers, are exhausted at a slow rate, decreasing as the difference of their squares increase. The square of 97, the greatest prime number of the first century, is 9,409 ; and the square of 199, the greatest in the second century, is 39,601. The greatest prime number in Brancker's Table is 313, whose square is 97,969. The following table exhibits the comparative numbers of prime numbers, in the successive thousands and tens of thousands of the natural series, from 1 to 100,000. A Table exhibiting the number of Prime Jfumbers by thousands and , tens of thousands. 1 2 3 4 5 -6 7 8 9 10 Sums 1 167 106 98 95 88 89 88 98 88 89 1006 2 135 103 104 92 101 97 87 95 92 85 991 3 127 109 100 106 102 89 88 90 89 97 997 4 120 105 104 100 85 92 93 83 84 86 952 5 119 102 94 94 96 90 80 92 87 87 941 6 114 108 98 92 86 93 98 91 85 95 960 7 117 98 101 99 90 99 84 83 ■ 88 84 943 8 107 104 94 94 95 91 99 95 93 82 954 9 110 94 98 90 89 90 80 84 76 87 898 10 112 104 92 96 98 930 94 81 91 94 87 949 Sums 1228 1033 983 958 924 878 902 876 879 9591 The whole number of prime numbers ending in 1, 3, 7 or 9, from 1 to 100,000, is The whele number of composite numbers of the same ter- 9,591 minations within the same limits, is 30,409 40,000 If to 9591 we add 2 for the prime numbers 2 and 5, we have the whole number of prime numbers from 1 to 100,000 9,593 Add the whole number of composite, 90,407 100^000 The average portion for each century is 9-593, or 9-j% nearly. The number is 9 in five successive centuries from the 803d to the 307th, inclusive. For some further remarks upon the relations of prime numbers see Appendix. PART II. OF THE METHODS DEVISED FOR THE DISCOVERY OF THE PRIME FACTORS OF COMPOSITE NUMBERS. CHAPTER I. OP THE CONSTRUCTION OP THE TABLE OF THE PRIME FACTORS OF THE ODD NUMBERS FROM 1 TO 30,000. The prime numbers having been discovered to the extent of one third of the highest limit proposed for finding the prime factors, and tables having been prepared similar to those made for the discovery of prime numbers, except that of making the spaces for the factors wider and reducing their number from 1,000 to 500 on a page, (see the tables of prime factors annexed), the work to be performed is similar in kind to that for finding the composite numbers, but far greater in extent than that. The essential points of difference are three. 1. Each one of the prime numbers must be inserted at the space of the lowest number divisible by it, next above that of the prime number itself, and in the space of every succeeding number so divisi- ble throughout the tables ; the passage from one page to another being continuous in the progressive order of the numbers. The common difference between the numbers in the table progres- sively, divisible by any prime number, is twice the prime number itsejLf, as was remarked in the rules for constructing the table for the discovery of prime numbers. 2. The process of inserting prime numbers must be continued until a prime number is reached that exceeds one-third of the highest number in the table, when it must be terminated. For example, if the table were limited to the number 499, such prime number would be the first in order that exceeds one-third of that number, viz. 167. Accordingly the highest prime number in- serted in such a table would be the next less on the list of prime numbers, which is 163. See for illustration the first page of the ta- bles of the prime factors of odd numbers^ 21 22 DISCOVERY OF PRIME FACTORS. If the table extend to 999, the first of the prime numbers excluded is 337; and the next less is the highest or last inserted. See the second page of the same tables. The highest prime number, therefore, to be found among the prime factors of the tables of odd numbers in this book, is 6661 ; the great- est of the factors of the number 19,983. AU the prime factors of all the odd composite numbers from 1 to 19999, embrace only the prime numbers from 3 to 6661 inclusive, being no more than eight hundred and ffty-eight. It is by the varipus combinations of these numbers, that all the composite numbers in these tables are constituted. 3. After all the requisite prime numbers are thus inserted in their places, the proper indices must be annexed to such of them as ought to be made powers. If in any space there be but one number, it is some root of its composite ; and its index will be found by involving it to a power equal to the composite number appertaining to its space. If in any space there are two or more numbers whose product is less than the composite number of their space, some one or more of them must have a proper index annexed ; so that when raised to the power which such index denotes, the product of the powers and of the factors that require no index, if any, will be equal to the com- posite number of that space. The requisite indices being annexed, the tables are complete. CHAPTER II, OF THE METHODS USED FOR CONSTRUCTING THE TABLES OP THE PRIME FACTORS OF THE EVEN NUMBERS FROM 1 TO 12,500. The forms of the tables for the prime factors of the even numbers are similar to those for the factors of the odd numbers, in all respects, except that the numbers in the left margin of each page are even numbers instead of odd ones. See the tables annexed. As the number 2 is the only even prime number, it or some of its powers, is one of the factors of every other even number. Those numbers, which zxepure oi perfect powers of 2, have no odd numbers among their prime factors. Exclusive of 2 itself, (which is called its first power) there are, within the scope of the table from 1 to 12,500, only twelve perfect powers of 2; viz., those from 2' to 2", inclusive. DISCOVERY OF PRIME FACTORS. 23 Every other number of these tables will be found to have some odd number or odd numbers, as well as the number 2, for its prime factor or factors. Upon examination it is found that every alternate term of the entire series of the even numbers, after 2, has 2 itself, that is, its first pow- er, and. not any of its higher powers, as a prime factor. This may be demonstrated in two ways, as follows: If each of the terms of the entire series of the odd numbers 1, 3, 5, 7, 9, &c. be multiplied by 2, as a constant factor, the products form the series 2, 6, 10, 14, 18, &c. to 98, in the first century of numbers ; the series 102, 106, 110, 114, 118, &c. to 198, in the second century; and so on, changing only the digits above those of the units and tens' places in the progression. Whence it appears, that every even number which terminates in any one of the terms of this series, 02, 06, 10, 14, 18, &c. to 98, in each century, has the first, and no higher, power of 2, as one of its iuctors. The same fact maj'' appear from the quotients arising from the di- vision of every term of the entire series of the even numbers 2, 4, 6, 8, 10, 12, 14, 16, 18, &c. by 2. The quotients are 1, 2, 3, 4, 5, 6, 7, 8, 9, &c. From which it is evident, that the number 2 is a factor but once, or in its first power only, of each of the terms of the series 2, 6, 10, 14, 18, &c. ; that is, 6f every alternate term after 2, of the entire series of the even num- bers ; while it is a factor in some of its higher powers, of all the other even numbers, viz. 4, 8, 12, 16, 20, &c. The first process in constructing these tables, therefore, is to write the number 2 in the first column opposite the number 6, and every alternate term thereafter, as 10, 14, 18, &c. throughout the tables. In every century after the first, these terms fall at the first, third, fifth, he. spaces, in every column. To understand how the other prime factors of this series are ascer- tained, we may consider the fact, already noted, that the successive terms of this series may be produced by the factor 2, and each of the terms of the series of the odd numbers. But the terms of this last mentioned series are comprised in the tables of the prime factors of the odd numbers. Such of them as are prime numbers are denoted by the blank spaces with dotted lines ; and such as are composite have their prime factors therein expressed. Nothing more is necessary, therefore, to supply, in the tables of the even numbers, all the odd prime factors of the terms 6, 10, 14, 18, &c. than to transfer to their proper spaces therein, from the table of 24 DISCOVERY OP PRIME FACTORS. odd numbers, every prime number and the prime factors of every composite number in successive order, inserting first the prime num- ber 3, opposite to the number 6, after its factor 2; then 5 opposite to 10; 7 opposite to 14; 3= opposite to 18; 11 opposite to 23; &c., to the end of the tables. Thus will all the prime factors of the series 6, 10, 14, &c. have been obtained. The prime factors of the other half of the even num- bers, comprised in the series 4, 8, 12, 16, 20, 8ic. may be as easily obtained. Having found and inserted in the table the prime factors of a few only of the terms of this series, for instance, from 4 to 48, as 2^; 2'; 2°-3; 2"; 2'-5, &c., all the prime factors of the residue of the series from 52 to 12,500 may be obtained from this table itself in the pro- gress of its construction. For, the prime factors of 52, 56, 60, and every succeeding term, will be the same as those of their halves, 26, 28, and 30, respectively, with this difference only, viz. : the index of the factor 2 as found in each of the halves, must be raised or increased by the addition or in- crement of 1. For examples : the respective prime factors of 26, 28, 30, are 2-13; 2'-7; 2-3-5; those of 52, 56, 60, are 2''-13; 3»-7; 2»-3-5. The prime factors of 102 are 2-3-17; those of 204 are 2''-3-17. So generally, the prime factors of any one of the terms of the series 4, 8, 12, 16, &c. throughout the tables, and to any greater extent, wiU be the same as those of the number equal to one half of it, except that its factor 2, must have an index greater by 1. By this simple rule, therefore, all the subsequent factors will be as- certained and inserted in the table, until it shall have been completed. The greatest prime factor in this table is 6247, which is the great- est prime number below 6500, and the half of the number 12494. If the table were extended to 20,000, the greatest prime factor included in it would be 9973, the greatest prime number below 10,000, and the half of the number 19,946. PART III. OF THE USE OF THE TABLES IN FINDING THE PRIME FACTORS OF NUMBERS WITHIN AND BEYOND THEIR LIMITS. CHAPTER I. OF THE MEANS OP FINDING THE PRIME FACTORS OP NUMBERS WITHIN THE LIMITS OF THE TABLES, THAT IS, FROM 1 TO 100,000. SECTION I.— OF ODD NUMBERS. Class I. — Of Odd Numbers from 1 to 20,000. All the prime factors of every odd composite number below 20,000 are contained in the author's table, entitled Prime Factors of Odd Numbers, which extends from page 96 to 135, inclusive. To find the factors of a given number in this table, observe first, within what century the given number is comprised, and then find, upon some page of the table, the particular column of factors for the century; and find also, in the left margin of the page, the line upon which are express- ed the digits of the tens and units of the given number. The prime fac- tors sought will be found in that column, upon that line, at their angular space. The prime factors of 17,931, for instance, are found on page 131, in the last colilmn, on the line of the marginal number 31. They are 3'43'139. Class II.— Op Odd Numbers from 20,000 to 100,000. Problem I. — To find the factors of any number ending in 5, vnthin the limits of this class. Rule. — Divide the given number by 5; then the divisor 5, and the quotient, if it be prime, or the divisor and the factors of the quotient, if it be composite, will constitute the required factors. N. B. As the quotient will always be less than 20,000, it and its fac- tors, when it is a composite number, may be found in the table of the factors of odd numbers. 4 25 26 DISCOVERY OP PRIME FACTORS. EXAMPLES. 1. What are the prime factors of 87,545 ? 87545 -r- 5 = 17509, a prime number. Am. 5-17509. 2. What are the prime factors of 87,537 ? 87537 -T- 5 = 17537 = 1319-71. Am. 513-1971. Problem II. — To find the factors of any composite number ending in 1, 3, 7 w 9, letween 20,000 and 100,000. To solve this Problem, Brancker's table is to be used in conjunction with the author's. That the reader may well understand the rule to be given for the solu- tion of this Problem and perceive the advantage gained in point of time and labor by using the tables together, an explanation will be here given of the manner in which Brancker's table or any other, which contains but the least one of the factors of each composite number, is to be used for finding the other factor or factors. Brancker's table does not contain the factors of numbers ending in 5; they being known to be divisible by 5, by their termination. All ■ the factors of numbers comprised in this entire table, might be found, without the aid of any other table, in the manner following. The three left hand digits or figures of any given number, being its thousands and hundreds, are to be found at the top of their proper column in the table, and the two right hand ones, being its tens and units, are to be found in the left margin of the page of that column. The least factor of the given number, if composite, will be in that column and on the line of the marginal digits. If the given number be prime, its character is marked by three dots in a line across the column. The least factor of the number 91,919, for instance, is found on page 201, in the last column, on the line of the marginal number 19. Having thus found in the table the least factor of a given composite number, observe whether it be in a smaller type and have a black line or dash under it or not; if it has, it is the square root of the given composite number. But if the factor found be not so distinguished^ the given num- ber must be divided by it ; and the resulting quotient is, in like manner, to be found lower down in the table, that is, nearer to its beginning. Then if the quotient be a prime number, the factor used as a divisor, and the quotient, will constitute the required factors. But if the quotient be a composite number, it must also be divided by its least factor which is set in the table. The second quotient is now to be sought still lower down in the table. If this be a prime number, then the two fetors used as divisors, and the second quotient, will constitute all the prime factors of the given number; but if it be composite, another division becomes necessary; and perhaps DISCOVERY OF PRIME FACTORS. 27 even a fourth. In every case, divisions of the quotients must be succes- sively made by their respective factors found in the table, until the last quotient is a prime number; then all the divisors used and such last quo- tient vifill be all the required factors. The number of the requisite divisions will always be equal to the number of the required factors, less one. It will be seen, however, that by incorporating the use of that part of Brancker's table which is above 20,000, with the author's, no more than one division will be necessary for finding all the prime factors of any composite number ending in 1, 3, 7 or 9, between 20,000 and 100,000 ; except as to numbers divisible by 3, between 60,000 and 100,000. For, if any odd number below 100,000, be divided by any prime num- ber greater than 3, the quotient will be less than 20,000, and within the ■author's table of factors of odd numbers. It may be observed in regard to all numbers divisible by 3, that al- though this number may always be found expressed in Brancker's table as their least factor or one of their least factors, yet it will never be ne- cessary to inspect this table to ascertain it; because it may be always discovered by an inspection of the given number itself. For if the sum of the digits of any number, or of all its digits, except 3, 6 and 9, be di- visible by 3, the whole number is so divisible, and may be divided at once without resort to the table. And if upon division of any number by 3, the quotient be less than 20,000, the use of Brancker's table may be T,^^,q„n+ 1 9. s i .*; 3 n « /i 9. fl fi s 3 38 THE USE OF PRIME FACTORS. The method of multiplying by factors may weU be practised by students in Arithmetic, as a pleasing variety in the mode of their work, and a profitable exercise in the use of numbers. It may be adopted by way oi proof oi work done in the ordinary way. It may be applied also in proof of the work of Division. For, as the product of the divisor by the quotient is equal to the dividend, less the remainder, if there be one ; it follows that the pro- duct of the divisor by the prime factors of the quotient, (omitting the fractional part, if any) or that of the quotient by the prime factors of the divisor, will be equal to the dividend less the remainder. For the method of multiplying fractions, by means of prime fac- tors, the reader is referred to Chapter IV, Problem 3 of this Part. OF DIVISION. To write prime factors to indicate the division of one number by another; or the division oj- the factors of one or more numbers by another, or by other numbers. RULES. I. When the dividend and the divisor are single numbers, write the factors of the dividend, or the dividend itself if a prime number, as the numerator, and those of the divisor, or the divisor itself if a prime number, as the denominator, of a vulgar fraction. II. When the dividend or the divisor is the unascertained product of two or more given numbers, write the factors of these numbers if composite, or the numbers themselves if prime, for the numerator or denominator, respectively, according to the rules given for writing the factors of a product ; uniting the like factors in each term of the fraction. The fraction thus constituted of factors will represent, and be equivalent to, the quotient to be ascertained when required. III. When in the numerator and denominator there are like or common factors, irrespective of their indices, place, Els nearly as prac- ticable, those of the denominator directly under those of the numera- tor, respectively ; as such collocation facilitates the cancelling of the common factors when it may be required. To obtain the quotient from the prime factors of the dividend and those of the divisor, arranged according to the preceding rules. RULES. I. Cancel or reject the common factors of the numerator and the denominator. THE USE OF PRIME FACTORS. 39 Factors whose roots or first powers are equal maybe considered common, notwithstanding they have unequal indices. When their indices are equal, both factors with their indices are to be cancelled; but when their indices are unequal, only that factor which has the less index is to be cancelled, either in the numerator or the denominator; while that which has, the greater must be retain- ed with a new index equal to the difference between the greater and the less of the two unequal indices. In truth so much only of the higher power is common as is equal to the lower. II. If after cancelling the common factors there be no factor left •either in the numerator or denominator, the number 1 must be sup- plied for a numerator or denominator. III. If after cancelling as above directed, the numerator be 1, the fraction is the true quotient sought. If the denominator be 1, it may be rejected, and the numerator will be the quotient. In all other cases, divide by the common rules for Division, or oth- erwise, the remaining factor, or the product of the remaining factors, of the numerator, by the remaining factor, or the product of the re- maining factors, of the denominator; and the quotient wiU be that which is sought. EXAMPLE S. 2-3 1 1. What is the quotient of o.o.k ■ -^ns. -g- 5-7-1M3 13 2. What is the quotient of q„ ^| — .? ^ns. -j-=13 2^-3=-7-31 3. What is the quotient of n-s-gs • 2^-3'-7-31 _ 2-3-7-31 _ 1302 54 ^ 2-3-83~~ 83 ~~ 83 •^^^' ^^83 N. B. Other examples of the cancelling of factors may be seen under the Rules ^ Problem 2, Chapter IV. To perform Division by means of factors, otherwise than by cancel- ling prime factors. Case I. — When the divisor is a composite number within one of the tables of factors, or a number whose factors are known. EuLE. — Divide first by one of the factors,' or by the product or power of two or more, and the quotient by another, and so on until all the factors of the divisor shall have been used. 40 THE USE OF PRIME FACTORS. EXAMPLES. 1. Divide 3456789 by 147. The factors of the divisor being 3-7«, the work may be done by short division, as follows : 3 )3456789 7 )1152263 7)164609 2 3 5 1 5f , the quotient. Remark. — Had the method of long division been used, the re- mainder would have been 84; which is four-sevenths of 147, the given divisor. The number 84 := 4'3'7. These factors are the remainder in the work as performed, and the first and second divisors used. In another example a remainder might occur at the end of more than one quotient, or indeed at the end of every quotient obtained. It is therefore important to know how to estimate the value of all the remainders that may occur in any case. It is evident that a remain- der at the end of the first quotient will be units of the dividend ; at the end of the second quotient, such part of units of the dividend as is denoted by the first divisor ; at the end of the third quotient such part as would be denoted by the product of the first and second divi- sor, &c., wherefore. To obtain the value of all the remainders Multiply the remainder at the end of the second quotient by the first divisor, and every subsequent remainder by the product of all the divisors preceding its own; and the sum of the products plus the remainder at the end of the first quotient, if any, will be the true re- mainder. 2. Divide 87654323 by 1155= 3-5-7.11. 3)87654323 5 )29218107 . .. .2rem. 7 )5843621 . ...2 " 11 )834803 75891. ...2 " The first remainder is .... 2 " second " 2'3= 6 " last " 2-3-5-7 = 210_ 218 ^ns. tSSdl^-g. Case II. — When the divisor is a prime number greater than 11 and less than 6661, and the dividend even or odd, composite or prime, is less than 20,000. THE USE OF PRIME FACTORS. 41 RULES. I. Find the prime factors of the dividend. If the divisor be one of them, the other of them or the product of the others of them, will be the quotient; and there' will be no re- mainder. II. But if the divisor be not a factor of the dividend, find the next less number, in the table of odd numbers, of which the divisor is a factor, and its co-factor, or the product of its co-factors, will be the integral part of the quotient sought, or such quotient less 1. III. Subtract the multiple of the divisor found from the dividend. If the remainder be less than the divisor, the quotient already found is the integral part of the quotient sought, and the remainder placed over the divisor will be the fractional part. But if the remainder be greater than the divisor, subtract the divisor from it and add 1 to the quotient found ; and the results will shew the true quotient. EXAMPLES. 1. Divide 9975 by 19. The factors of the dividend are 3-5'-7-19. Ans. 3-5'-7 = 525. 2. Divide 9973 by 23. The dividend is not a multiple of the divisor. But 9959 is a mul- tiple, and its factors are 23433; and 9973 — 9959 = 14 the remain- der, ^ns. 4331A. 3. Divide 19999 by 53. 19981 is a multiple of the divisor, and its co-factors are 13-29:=377. Ans. 377if . Remarks. — Every prime number, from 3 to 6661, is included among the prime factors oV odd numbers. If its magnitude do not exceed 50, it occurs once, at the least, in each column of the factors; if it be between 50 and 100 it occurs once in the range of two co- lumns ; between 100 and 150, once in the range of three columns, &c. The place of any factor therefore not exceeding 500 may be readily found, by running the eye up or down the columns. But as divisors of greater magnitude would require a longer search, every practitioner will judge for himself when the rules of this case may be advantageously applied, either for performing the work of division or for proving it. As the product of any prime number by any number whatever is a multiple of the prime number, it is easy to find such a multiple. And then any desired multiple may be found by means of the common difierence of its successive multiples, which for all the numbers, odd 6 42 THE USE OF PRIME FACTORS. and even, of the natural series, is the prime number itself; but which for the odd numbers of the table, is twice the prime number. If the multiplier assumed be an even number, the product will also be an even number. If to such product there be added, or from it sub- tracted, the prime number itself, or the product of it and some odd number, a multiple will be obtained among the odd numbers. Sup- pose, for instance, it were required to find the greatest multiple of the divisor 991, in the dividend 19947. Multiplying the divisor by 10, we obtain the multiple 9910. Doubling this product we have 19820, the multiple sought. If now we desire to find, in the table, the factors of the greatest multiple therein, next less than the divi- dend, we subtract the divisor from the multiple found, and obtain 18829, the multiple in the table next less to the dividend. This shews the reason of having sometimes to add 1 to the quotient first found by the Rules of this Case. Though this process of finding multiples differs not in principle from that of common division, yet, by the aid of the Rules of this Case and those of the next, the work of division may be sometimes much abbreviated, especially when the quotient is a large number. Case III. — When the divisor is a prime number greater than 11 and not greater than 6661, and the dividend even or odd, composite or prime, exceeds 20,000. RULES. I. Find, by the Rules of Case II, or otherwise, the greatest mul- tiple of the divisor, in the table, or any multiple less than the divi- dend, whose factors may be found, and find and set down the quotient of such multiple divided by the divisor, as & partial quotient. II. Multiply both the multiple and the partial quotient found, by any number that will give a product less than the dividend, but as near to it in magnitude as possible ; and the products call respective- ly ^. proximate multiple and ^ proximate quotient. Or if the first mul- tiple be more than half of the dividend, add to the multiple found such other multiple of the divisor as will make a multiple proximate to the dividend, and add also to the partial quotient the quotient of this multiple divided by the divisor, and call the sums respectively a proximate multiple and proximate quotient. III. Subtract the proximate multiple from the whole dividend, and by any method at option, divide the remainder by the divisor. Add the quotient obtained by this last process to the proximate quo- tient, and the sum will be the quotient sought. THE USE OF PRIME FACTORS. 43 EXAMPLES. 1. Divide 999,545 by 47. The greatest multiple, in the table, of 47, is, 19975, and the co- factors of 47 are 5'- 17 = 425, the partial quotient. Multiplying both the multiple found and the partial quotient by 50, or J-|i, we obtain a proximate multiple, 998750, and a proximate quotient, 21250. Subtracting now the proximate multiple from the dividend, we have the remainder 795, still undivided. This being divided by 47, by the Eules of Case II, gives a quotient of 16|5, which added to the proxi- mate quotient gives the quotient sought. ^ns. 21266|f . For the division of fractions by means of factors, see Chapter IV, Problem 4, of this Part. CHAPTER II. OF THE MEASURES OF NUMBERS. DEFINITIONS AND EXPLANATIONS. A measure of a given number is a number that will divide it with- out a remainder. A common measure of two or more numbers is a number that will divide each of them without a remainder. The greatest common measure of two or more numbers is the great- est number that will divide each of them without a remainder. Every integral number may be measured by itself and by a unit. A prime number cannot be otherwise measured. Every composite number may be measured not only by itself and by a unit', but also by each one of its different prime factors, and by as many composite numbers as may be formed by different combina- tions of those factors ; it being understood that the product of all the prime factors is the same as the number itself. No number, which is a power of a prime number, can have any measures other than the prime number itself and numbers which are its powers. But a number which is a power of a composite number may have for its measures, the prime factors of the root and their different pro- ducts and powers. As for example, 6:=2'3, and 6'i=2'-3°, and 10 = 2-5, and 10'=:2'-5«, &c. 44 THE USE OP PRIME FACTORS. To render the subjects of measures and multiples the more easily intelligible, the following table has been devised. A TABLE DESIGNED TO EXHIBIT MEASURES AND MULTIPLES. 1 2 3 4 5 6 7 8 9 10 11 12 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 .18 19 20 21 22 23 24 2 'V 3 4 5 6 7 8 9 10 11 12 12 2 3 2 "2" ' V 4 5 2 2 ' 2' 2 2 "2' 2 3 4 6 7 8 9 10 11 3 3 "V " 4' 4 5 S 6 7 3 4 • • ■ 6 8 Explanations of measures with reference to the foregoing Table. AU the numljers on each line, except that constituting the head of the table, are, with the exception of a unitj all the possible mea- sures of the first or greatest one of them. Neither of the prime numbers 2, 3, 5, 7, &c. has any measure but itself. Each of the composite numbers in the natural series, in the left margin of the table, has two or more measures. The numbers 2 and 4 are measures of 4 ; and the numbers 2, 3 and 6 are measures of 6. The numbers 4, 8, 16 being powers of 3, have no other measures than 2, and one or more of its powers. The table is not extensive enough to shew the same truth in regard to 3, and the larger prime numbers. Other examples may be seen in the table of powers and in the tables of prime factors. THE USE OF PRIME FACTORS 45 Each of the numbers repeated iij the several columns is a common measure of all the composite numbers in the left margin of the table on whose lines it is set dowa, The number 2 is a common measure of all the even numbers, 3 of each of the terms of the series 3, 6, 9, 12, 15, &c., and 4 of each of the terms of the series 4, 8, 12, 16, 20, &c. And, generally, if any number of the natural series be made the first term and also the com- mon difference of a series in Arithmetical Progression, such number will be a common measure of aU the terms of the series to any ex- tent, ad infinitum,; and it cannot be a common measure of any number not embraced in such series. If several series, so formed for different numbers, be compared to- gether, it will be found that certain terms in one series are respec- tively equal to certain terms in each of the others ; and that each of the common measures of the terms of each series will be a common measure also of all such terms of each of the other series as are con- tained in its own. For example : the series for the measures 2, 3 and 4, respectively, may be written so that each series after the first shall contain only such numbers as are found in the first ;• thus — (2) 3 4 6 8 10 12 14 16 18 20 22 24 (3) 6 12 18 24 (4) 12 24 Whence it appears that 2 and 3 are both common measures of 6, 12, 18, and 24 ; while 2, 3, and 4 are all common measures of only 12 and 24. To write a series, therefore, such that certain given numbers shall be common measures of all its terms, it is only necessary to make their product the first term and also the common difference of the series. A knowledge of the common measures of different numbers, or of their greatest common measure, which is the product of all their common measures, is necessary in order to reduce fractions and ratios to their lowest terms ; and rules have been laid down by arithmeti- cians for finding them. It is here proposed, by the use of the tables of prime numbers and of prime factors, to find them by comparatively short methods. To avoid, as far as possibliB, the useless labor of applying rules for the purpose of finding the common measures of numbers which have no common measure, a few criteria will be laid down for determining whether given numbers have any common measure or not, before any rule is given for finding the greatest common measure of numbers. 46 THE USE OF PRIME FACTORS. CRITERIA OR MEANS FOR DETERMINING WHETHER GIVEN NUMBERS HAVE ANY COMMON MEASURE OR NOT. I. If all the given numbers are even numbers, they have, at the least, one common measure, viz. 3. II. If the sum of the digits of each of the given numbers is divisi- ble by 3, the numbers have, at the least, one common measure, viz.' 3. III. If all the given numbers end in 5, or some of them in 5, and the others in 0, they have, at the least, one common measure, viz. 5. IV. If two or more of the given numbers are unequal or different prime numbers, the given numbers have no common measure. V. If only one of the given numbers is a prime number, they cannot have a common measure other than the prime number itself; which cannot be their common measure unless it be the least of all of them and be moreover a factor of each of them, exclusive of itself. VI. If the given numbers are composite odd numbers not aU end- ing in 5, or if some of them are odd and the others even, they cannot have a common measure, unless they have, at the least, one common prime factor. In the application of the three last of the foregoing propositions the tables may be used. PROBLEM 1. To find the greatest common measure of numbers that have a com- mon measure or common measures. RULES. I. If one of the given numbers be a prime number, it will be their greatest common measure, if it be a common factor of the others. II. If all the given numbers be composite and have but one com- mon prime factor, it wiU be their greatest common measure. III. If all the given numbers be composite and have tvx> or more common prime factors, the product of all such factors, of any one of the given numbers, will be their greatest common measure. N. B. Of factors of the same root and egual powers, each is a common factor; but of factors of the same root and unequal powers, that of the least power only is a common factor. EXAMPLES. 1. What is the greatest common measure of 3, 4, 6, 8, 10? ^ns. 3. 3. What is the greatest common measure of 359, 469, 731, 833, «73? ^«,. 7. THE USE OF PRIME FACTORS. 47 3. What is the greatest common measure of 5117, 6511, 7939, 8823? ^ns. 17. 4. What is the greatest common measure of 123, 287, 369, 451, 533, 697? Ans. 41. 5. What is the greatest common measure of 16, 64, 512. 4096? Each of the numbers is a power of 2. Ans. 2'' =: 16. 6. What is the greatest common measure of 27, 81, 243, 729, 2187 ? Each of them is a power of 3. Ans. 3' = 27. 7. What is the greatest common measure of 36, 216, 1296, 7776? Each of them is a power of 6. Ans. 2'-3* =z 6" = 36. 8. What is the greatest common measure of 125, 525, 6775, 8975? Ans. 5=^ = 25. 9. What is the greatest common measure of 15, 135, 465, 585, 795? Ans. 3-5 = 15. 10. What is the greatest common measure of 627, 1881, 6897? Ans. 3- 11- 19 = 627. 11. What is the greatest common measure of 19747 and 19943 ? Ans. 7^=49. 12. What is the greatest common measure of 11111111, 22222222 and 33333333? Ans. ll-73-iOM37= 11111111. For the methods of reducing fractions and ratios by means of the common measures, or the factors of their terms, see Chapter IV, Problem 2. PROBLEM 2. To find, from its prime factors, all the composite measures of any composite number in the tables. Case I. — When the given number is a power of a single prime number. RULES. Involve or raise the prime number to the power denoted by its in- dex, and then all the powers obtained, from the second to the last inclusive, will be the measures sought. Or, divide the given number by the prime number and the first and every successive quotient in like manner, until a quotient is ob- tained equal to the divisor. Then the given number and all the quo- tients, exclusive of the last, will be the composite measures sought. N. B. The number of the composite measures of any power of a prime number is equal to the index of the prime number, expressive of the power, less 1. 48 THE USE OF PRIME FACTORS. EXAMPLES. What are the composite measures of 64 = 2° and of 729 = 3°, re- spectively ? 2-2=2'= 4 2)64 3-3 = 9 3)729 2^.2 _ 23=: 8 2)32 3'-3= 27 3)243 2=-2=2* = 16 2)16 3'-3= 81 3)81 2^-2 = 2' = 32 2)8 3*-3 = 243 3)27 2'-2 = 2' = 64 2)4 2 3'-3 = 729 3)9 3 Jlns. 4, 8, 16, 32, 64, measures of 2°, •■ and 9, 27, 81, 243, 729, of 3'. Case II. — When the given number is the product of two ormore different prime numbers. RULES. I. If the given number have but two prime factors, their product, or the given number itself, is its only composite measure. II. If the given number have three prime factors, it will have/ow composite measures, viz., itself, or the product of all of them, and the three different products that can be made of them by taking two at a time. III. If the given number have four prime factors, it will have eleven composite measures, viz., itself, and six products of two factors each, and four products of three factors each. IV. If the given number have Jive prime factors, it wiU have twenty-six composite measures, viz., itself, ten of two factors, ten. of three, and five of four factors each. There is no number in the tables that has more than five different prime factors. N. B. To obtain the products, the factors are to be combined, in accordance with the laws and rules laid down by arithmeticians, for finding all the combinations that can be made of a given number of different things by taking a given number of them at a time. EXAMPLES. 1. Required all the composite measures of 30 = 2-3-5. Let the factors be represented by the letters a b c 2- 3- 5 The different combinations and products, in addition to the given number itself, are as follows : ai = 2-3= 6 ae = 2-5 = 10 i c = 3-5 = 15 Jlns. 6, 10, 15, 30. THE USE OF PRIME FACTORS. 49 2. Required all the composite measures of 210 = 2-3'5-7. Let the factors be represented by the letters abed. 2-3-5-7 COMBINATIONS AND PRODUCTS. ffi5=2-3=6 abc = 2-3'5 = 30 oc=2-5=10 a6rf=2-3-7= 42 «rf=2-7=14 «crf=2-5-7=z 70 b c = 3-5 =: 15 b cd= 3-5-7 = 105 bdz=3-7=21 cd = 5-7 = 35 Ans. 6, 10, 14, 15, 21, 8CI, 35, 42, 70, 105, 210. 3. Required the composite measures of 2310 = 2-3-5-7'll. abode 2- 3 -5 ■ 7- 11 COMBINATIONS AND PRODUCTS. a 5 = 2-3 = 6 ffi6c = 2-3-5 = 30' o i c <^ = 2-3-5-7 = 210 ac=2-5 =10 abd=2-3-7 = 42 a 4 c e = 2-3-5-ll = 330 ad=2-7 =14 fflde = 2-3-ll= 66 a * rf e = 2-3-7-ll = 462 « 6 = 2-11 = 22 fflcc?= 2-5-7 = 70 a e rfe= 2-5-7-11= 770 bc = 3-S =15 fflce=2-5-ll = 110 6 c rf e = 3-5-7-11 = 1155 bd=3-7 =21 orfe = 2-7-11 = 154 6 6=3-11 = 33 bcd=3-5-7 =105 cd= 5-7 =35 6ce=3-5-ll = 165 c 6 = 5-11 = 55 i«/ 6 = 3-7-11 = 231 de = 7-n = 77 crf6=5-7-ll = 385 These numbers may be arranged in the order of their magnitudes, and the given number itself be added, for the measures required. . Case III. — When some or all of the prime factors of the given number have indices. RULES. I. In lieu of the factors with indices, write equivalent factors with- out indices, by expressing each factor as many times as there are units in its index. II. Make the combinations and products of these factors in the same manner as those were made in the examples of the preceding Case, rejecting in the progress, or cancelling on the completion, of the work, aU but one of those which are of the same combination. EXAMPLES. 1. What are the composite measures of 45 = 3^-5 .? a a b a az::^ 3-3 zr 9 3 ■ 3 ■ 5 a Zi = 3-5 = 15 ab~t-i=:U Jlns. 9, 15, 45. 50 THE USE OF PRIME FACTORS. 2. What are the composite measures of 225 = 3*'5*? a a b b 3-3-5-5 ao = 3-3 =9 aab = 3-3-5 = 45 a b = 3-5= 15 aab= $■$:$ = )i$ ab=z$-$=:X.$ ab b = 3-5-5 = 75 a b = $■$:=: U ,abb = %-H = t$ a b =z $•$ ^ X$ bb=5-5z=25 Ans. 9, 15, 25, 45, 75, 225. 3. What are the composite measures of 315 = 3'''5'7? a a b c 3-3-5-7 00 = 3-3=9 aab = 3-3-5 = 45 o 6 = 3-5 = 15 aac = 3-3-7 = 63 o c = 3-7 = 21 abc = 3-5-7 = 105 a b = $•$ = i$ abc =.-%•$•% ■= 10$ ac = $-t = U 6 c = 5-7 = 35 Ans. 9, 15, 21, 35, 45, 63, 105, 315. Remarks. — The foregoing Rules, with their examples, suffice to shew the principles upon which the composite measures of any num- ber may be obtained. They are the more numerous in proportion to the number of the prime factors and the magnitude of their indicies. In establishing systems of weights and measures, Governments have had regard to the number of aliquot parts, or integral measures, which are contained in particular numbers, with the view of obtain- ing many subdivisions without fractions. For instance, 112 lbs. avoirdupois weight is a cwt., now sometimes called the long hundred, to distinguish it from 100, which is called the short hundred. It may be seen from the prime factors of 112, which are 2''7, that it has many different measures ; so too the number 960 = 2°-3-5, which is the number of farthings in a pound sterling, has a very great number of measures. Even numbers of the series 04, 08, 12, &c., are the only numbers that admit of many subdivisions. These numbers always contain the second or some higher power of 2. Its index in the table of factors will shew how many subdivisions may be made of a given number by dividing it by 2, and the first and each successive quotient by 2 without descending to fractions. If its index be 1, the given number can be divided into halves only, if its index be 2, into halves and quarters; if 3, into halves, quarters and eighths, &c. Hence 112 is divisible into halves, quarters, eighths and sixteenths, THE USE OP PRIME FACTORS. 51 and 960 into all those parts and two smaller ones. All the measures: of the former number are even, except the number 7; and all of the- latter, except 3, 5 and 15. Every composite measure which has the number 2 among its fac- tors is an even number. Those numbers, 112 and 960, have likewise- among their measures the powers of 2, as well as the different pro~ ducts that may be formed of them and their other prime factor ox factors. CHAPTER III. OP THE MULTIPLES OF NUMBERS. DEFINITIONS AND EXPLANATIONS. A multiple of a given number is a number that contains the given number some number of times exactly; so that when divided by it there is no remainder. Or a multiple of a given number is a number that may be exactly measured by the given number. Multiples, therefore, may, be produced by Multiplication and tested by Division. If every number of the natural series, except the first, be multiplied by any number, as a constant factor, each of the products will be a multiple of such constant factor. Every product, except the first, in each line of the Multiplication Table, for instance, is a multiple of the number used as a multiplier in producing it. The multiples of any particular number may be written in a series to any extent, by making the particular number the first term and also the common difference of the series. For all the terms, except the first, of such a series will be multiples of the first term or of the common difierence. If any multiple of any particular number be made the first term, and the particular number itself the common difference of a series, every term of the series will be a multiple of the particular number or common difference, but only some of the terms will be multiples of the first. The multiples of 13, for example, may be written thus: 13, 26, 39, 52, 65, 78, 91, &c. The multiples terminating in any particular digit may be written in a series by assuming the least, or any multiple of that termination 5-2 THE USE OF PRIME FACTORS. for the first term, and ten times the particular number whose multi- ples are sought, for the common difference. To write the multiples of 13 terminating in 1, for instance, make 91 (which is equal to7'13) the first term, and 130 (which is equal to 10-13) the common differ- ence, and the required multiples will be as foUows : 91, 221, 351, 481, 611, 741, 871, 1001, &c. The number 13 being a prime number, it will be found, by refer- ence to the table of the prime factors of odd numbers, to be one of the factors of the above series to the number 19991, the last of its multiples, of that termination, within the limit of this table. The series may be continued ad infinitum. In like manner may be written the multiples, of any termination, of any particular number prime or composite. A prime number cannot be a multiple. But every composite num- ber is a multiple of certain numbers. A common multiple is a number that has two or more measures, by each of which it may be exactly measured, or by each of which it may be divided without a remainder. If the multiples of two or more given numbers be separately written in a series, those terms of each series, which are common to all the different series, will be the only common multiples therein, of the given numbers. ~" The comparison of the common or equal terms may be the more easily made, if, in every series after the first, the terms not common to it and the terms of the preceding series be omitted. For example, to ascertain the common multiples of the numbers 2, 3 and 4, wnte their multiples thus : Multiples of 2 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24. "3 6, 12, 18, 24. " "4 12, 24. From a comparison of the terms of the series it appears that 6, 12, 18, and 24, are common multiples of 2 and 3 ; while 12 and 24 only are common multiples of 2, 3" and 4. Common multiples of two or more numbers may be written in a series to any extent by making their product the first term and also the common difference of the series. The common multiples of 2, 3 and 5 are 30, 60, 90, 120, &c. Those of 7, 11 and 13 are 1001, 2002, 3003, 4004, 5005, 6006, 7007, 8008, 9009, &c. By reference to the tables of prime factors, the numbers 7, 11, 13, will be found among the prime factors of each of the numbers of this last series. THE USE OF PRIME FACTORS. 53 Every composite number is a common multiple of all its different measures, prime and composite, inclusive of itself. The least common multiple of two or more numbers is the least number that may be exactly measured by each of them. A number may be the least common multiple of itself and one or more other numbers. On reference to the table of measures and multiples, at page 44, it may be observed, that each of the composite numbers in the natiiral series, in the first column, is the least common multiple of all the numbers on the same line. As 4 is the least common mul- tiple of 2 and 4; 6 of 2, 3, and 6; 8 of 2, 4, and 8; 9 of 3 and 9; 10 of 2, 5 and 10 ; 12 of 2, 3, 4, 6, and 12, &c. The greatest common multiple of any number is indefinitely or in- finitely large and may be called an impossible number. The multiple which is most frequently required, for the solution of certain questions, is the least common multiple. This is necessary in order to reduce fractions to their least common denominator, to as- certain the cycles of time that are measured by the successive con- currence or conjunctions of bodies moving in circles or orbits of other forms, &c. To find the least common multiples of numbers, by any of the rules laid down in Arithmetic, generally requires a long and tedious opera- tion. By the aid of the tables of prime numbers and of prime factors they may be readily found. PROBLEM 1. To find the least common multiple of two or more numbers. RULES. I. If the numbers are all prime numbers their product wiU be their least common multiple. II. If the numbers are all composite numbers, arrange aU their prime factors on one line. If no two or more of the factors are like, that is, of the same magnitude without respect to their indices, if any, then the product of, the given numbers will be their least common multiple. But if, irrespective of their indices, any two or more of all the factors are like, cancel all but one of the like factors, retaining that of the highest power with its index. Then the product of all the uncancelled factors will be the least common multiple of the given numbers. III. If some of the numbers are prime and the others composite, arrange all the prime numbers and the prime factors of the composite numbers on a line as factors of one number, and then proceed by the directions of the next preceding rule. 54 TllE USE OF PRIME FACTORS. N. B. Instead of actually cancelling the supernumerary factors, it will generally be more convenient to pursue the following process : First. Select from all the prime numbers of the question and those of the composite numbers, one of every different magnitude, without regard to their indices, beginning with the least and ending with the greatest and set the numbers selected in one line. Secondly. Affix to each of the numbers thus selected the highest index which it has among all the factors of the composite numbers. Thirdly. Eaise each of the numbers that has an index to the power indicated by its index. Lastly. Find the product of all the numbers thus selected and prepared, and it will be the least common multiple of the given numbers. The rules will be rendered clearly intelligible and easy of practice by the following examples. EXAMPLES. 1. What is the least common multiple of the prime numbers 3, 3, 5, 7? ^ns. 210. 2. What is the least common multiple of the composite numbers 4, 18, 60, 72? Their prime factors are 2^ 2-3=, 2''-3-5, 2'-3». The factors to be selected, with their respective indices, are 2'"3''5, which are equivalent to 5-8-9, whose product, 360, is the least com- mon multiple sought. 3. What is the least common multiple of the nine digits.' Omitting the first as immaterial in the solution of the question, we have 2, 3, 5, 7, prime numbers, and 4, 6, 8, 9, composite. The prime factors of the composite numbers are 2^, 2*3, 2', 3^ Taking one of each of the different prime numbers of the question and of the factors, (which in this example are not different except in their indices,) we have the prime numbers of the question as the se- lected numbers ; and affixing to the numbers 2 and 3 respectively the highest index which each has among the factors, we have the prime factors of the multiple sought, viz. : 3'*3*'5'7, or their equivalents 5-7-8-9=:2520. Ans. 2520: 4. What is the least number of days that may be divided by the number of days in each of the twelve calendar months of a common year, without a remainder? All the different lengths of the months being but three, viz: 28, 30 and 31 days, it is evident that the least common multiple of these numbers will be the true answer to the question. The last of them THE USE OF PRIME FACTORS. 55 is a prime number. The respective prime factors of the other two are 2^-7, 2-3-5. The different numbers necessary to be retaJned with their indices, are 2''-3-5-7-31 = 13020 days Ans. N. B. The answer to a similar question, for the months of a com- mon year and of leap year also, would be 29' 13020 = 377580 days. 5. The hour, minute and second hand of a watch being together at 12 o'clock, when will they next be together again? The respective periods of their revolutions measured by minutes, in the reverse order, are 1, 60, 720. The prime factors of the com- posite numbers are 2^-3'5 and 2'''3^-5. Those of the answer are the last only which are equal to 720 min- utes or 12 hours. Ans. 12 o'clock. 6. There is an island 100 miles in circumference, and four men start together to travel around it in the same direction. A travels 20 miles a day, B 25, C 30, and D 35. How many days must they travel to come together again, and how many revolutions will each have made? >_o_o, i-ij), iJ_o, i_!LP :^-i_ i.^ i_o^ 2_o^ the respective numbers of days requisite for one revolution by each of the respective travelers. The least common multiple of the numerators of these fractions 2'''5 =: 20, is the number of days at the expiration of which they will all come together again. And Y. V. ¥ X tV and V X /b=4, 5, 6, and 7, the respec- tive numbers of their revolutions. 7. Three men, resident in one city, are in the habit of making regular journeys in certain periods of time, viz: A in 20, B in 30, and C in 40 days, each tarrying at home 2 days at the end of each journey. If they all commenced their travel on the first day of May 1851, what is the shortest period before they will all return to the city on the same day, and what will be the year, month, and day of the month, when they so return ? In order to embrace the time from the beginning of one journey to the beginning of the next succeeding one, 2 must be added to the length of each man's journey, making their respective times 22, 32, 42. The least common multiple of these is 2'-3-7-ll=:7392 days, which brings the beginning of some journey. So that they all reach home together in two days less time, or 7390 days ; which, allowing 365J days to a year, is equal to 20 years and eighty-five days. Am. They will return together on the 24th July, 1871. 56 THE USE OF PRIME FACTORS. PROBLEM 2. To reduce given fractions to others which shall have the least com- mon denominator. RULES. I. In lieu of the given fractions, write equivalent ones, by substi- tuting for every composite numerator and denominator, its prime fac- tors, retaining all the numerators and denominators which are prime numbers. II. Find, by the rules of Problem 1, the least common multiple of the given denominators, from their factors already expressed, and it will be the least common denominator. III. In lieu of the fractions written, write now those required or their equivalents, by substituting for the denominator of each fraction, the least common denominator found, and by adding or uniting 'to its numerator all the factors of the common denominator not contained among those of its given denominator. EXAMPLES. 1. Reduce \, f , f , J, ij and ij to equivalent fractions having the least common denominator. The numerators of all the given frac- tions and the denominator of the first of them being prime numbers, the equivalent fractions are — , — , , — , — , . ^ 3 2^ 2-3 2' 2^ 2'-3 The least common multiple of the denominators is 2*'3. The fractions equivalent to those required, are — _2^ 2^ 2^ 2;;3^ 341_ 2^ 2*-3' 2*-3' 2^-3' 2'-3' 2^-3' 2^-3 ' -^ns. i|, If, if, if, If and If. 2. What is the sum of i, \, \ and \ ? Their equivalents are , , , , and the least common ^ 3 2' 5 2.3 denominator is 2'-3-5=: 60. The respective numerators for it will be 20, 15, 12, 10. Ans. U = i!. 3. What is the sum of ^, |, |, A and f .? 1 o Q 02 g Their equivalents are , _, _, _, ___. Their least common 2 3 2'' 5 2-3 denominator is 2'-3-5 = 60. The numerators for this are 30, 40, 45. 48, 50. Am 30+40+45+48+50 _ 213 _ an. 60 ~ 60 ~ ^^ 4. What is the difference between ^f and f| } THE USE OF PRIME FACTORS. 57 2.32 2-11 The equivalents are - — , ; and the least common denomina- toris2'-3= = 288. - Jlns '^ — ^'^'^^ = 144 — 66 _ ^ _ 2-3-13 __ 13 a^-S" 2^-3= 288 288 2^ 48' 5. If from 17|, 13f be taken, what will remain ? 71 3-37 The equivalents are —- , — -— , and the least common denomina- 2 2 tor is 2^ = 8. 8 8 8 8° CHAPTER IV. OF RATIOS AND FRACTIONS. DEFINITIONS AND EXPLANATIONS. Ratio has respect to the relative magnitudes of numbers or quan- tities. The measure of a ratio between two numbers is the quotient arising from the division of one of them by the other. The quotient is not often absolutely expressed, but it is indicated or implied by some juxtaposition of the numbers whose ratio is in- tended to be expressed. The numbers are usually placed on a horizontal line with dots be- tween them thus 2 : 4, or 4 : 2 ; or placed one under the other with a line between them in the form of a fraction, thus, |, |, making the first or antecedent term of the ratio the denominator or divisor, and the second term or consequent the numerator or dividend. When the two numbers are equal the quotient of measure must always be 1. This quotient therefore indicates the ratio of equality. When the numbers, compared with reference to their ratio, are unequal in any degree, the quotient will always be greater or less than a unit, according as the greater number is divided by the less or the less by the greater; and the greater the inequality of the num- bers the greater or less will be the value of the quotient compared to a unit. Taking the terms of the fraction, instead of their absolute quotient, as virtually the measure of the ratio, it is easy to see that the same 8 58 THE USE OF. PRIME FACTORS. ratio may be expressed or indicated by any number of fractions, pro- vided they are all of equal value in comparison to a unit. If, for example, the terms of the fractions f and ^, be respectively multiplied by each of the successive numbers of the natural series above the first, as by 2, ' 3, 4, 5, 6, &c. the products will give new fractions all different in regard to the magnitudes of their terms, but all equal when measured by the quotient arising by a division of one term by the other of each of them, and each equal to the original factions, respectively ; as is evident from an inspection of the result ing fractions a, a, i, y, y, &c. and |, f, f, ^, jl, &c., each one of the former series being of the value of 2, and each of the latter of |. The ratio of each series is called the inverse of that of the other; because if the fractions of either one series be inverted, they become identical with those of the other. The same results would follow if the terms of either one of the new fractions were multiplied in like manner to any extent. And the same law would be observable in regard to any ratio, how- ever expressed. If each of its terms be multiplied by one and the same number of any magnitude whatever, the products of the terms by such number would give a fraction of the same value or ratio as that of the original. A decimal fraction in like manner expresses a ratio between its decimals which are its numerator, and the number which is' always understood or known to be its denominator, that is, a number equal to 1, with as many cyphers annexed as there are digits or places of decimals in the numerator. Circulating decimals, however, when converted into their equivalent vulgar fractions have denominators of less comparative magnitude. In Division the divisor and dividend always express a ratio, and may be written in the form of a fraction. As the ratio between any two numbers of small magnitude will be preserved or remain unchanged when each of its terms is multiplied by the same number ; so, it is evident, a ratio expressed in terms of great magnitude will be equally preserved when each of its terms is divided by the same number, provided each is divisible without a remainder. To avoid the long and tedious processes of using numbers of great magnitude in the solution of various problems in Arithmetic, espe- cially in Fractions and Proportions, it is often necessary to reduce fractions and ratios to what is called their lowest terms; that is, to re- duce their terms to numbers of the least possible magnitude whereby the measure or value of the original fractions or ratios may be ex- pressed. It is now proposed to shew how this may be expeditiously done by THE USE OF PRIME FACTORS. 59 the use of prime factors of composite numbers, taking advantage of the explanations and rules already given in regard to measures and multiples. PROBLEM 1. To determine whether the terms of a given fraction or ratio are re- ducible or not. RULES. I. If both the terms of the given fraction or ratio are prime num- bers they are not reducible. II. If one of the terms is a prime number and the other a com- posite one, they are not reducible unless the prime number be less than the composite and be also a factor of it. III. Though both of the terms are composite numbers, they are nevertheless not reducible, unless they have at least one common prime factor, that is, some common measure. PROBLEM 2. To reduce a fraction or ratio, which is reducible, to its lowest terms. RULES. I. Divide each of the terms of the given fraction or ratio, by their greatest common measure. II. Or divide each of them, first by one of their common prime factors and the quotient by another and so on, until each of the com- mon factors shall have been used as a divisor, when the last quotients wiU be the terms of the required fraction or product. III. Or, (what is the same thing in effect,) cancel the common prime factor or factors of the terms of the given fraction or ratio, then the remaining factor or factors, or their products, will be its lowest terms. Should all the factors of either term be cancelled, the num- ber 1 must be written for a term in their place. N. B. No greater number of factors of any one magnitude must ever be cancelled in either term of a ratio or fraction than what is cancelled in the other term ; for the cancelling of any one factor of a number is equivalent to a division of the number itself by such factor. If the common factors or their roots have different indices, that which is retained in its proper place in the numerator or denominator must have a new index equal to the difference of the indices of such factors. EXAMPLES. 1. Reduce ||i to its lowest terms. The factors are = — , the lowest terms. 211-31 2 60 THE USE OP PRIME FACTORS. Whenever, as in this example, it may be seen upon inspection, what part the nuitjerator is of the denominator, the use of the tables will be unnecessary. 2. Reduce i*f. 5-7-17 5-7 35 96 9 The factors are 3- 17- 19 3-19 57 3. Reduce ||4|. 7- 17- 19 _ 17-19 _ 323 3-7-l37~" 3-137 ~ 411" Reduce |f|f y-37 _ 3^37 _ 333 5. Reduce ffaf, 6. Reduce ^fff- 7. Reduce f^ff. 8. Reduce Uih 2'-ll-101 As, in this example, each of the digits of the numerator is to the corresponding digit of the denominator in the ratio of 7 to 8, the whole numerator must bear to the denominator the same ratio. 9. Reduce ^V*/^- 67-149 _ 149 3^383 383 ■ 383' 5-7-lP 7-11 77 5=-ll-17 5-17 85' 3-31-73 31 31 3^-5-73 ~ 3'-5 ~45' 3^-5'' 3' 27 3-5»-7-19 7-19 "183 7-11-101 7 2''-3=-7'-67 1764 10. Reduce m|f . 2'- 11-907 _ 907 2'-3-7-ll-43~1806" 11. Reduce the decimal .625 to its lowest terms in a vulgar frac- tion. _5^ _ ^ 2''-5^ ■" 8 ' 12. Reduce .555555. , 3-5-7-1M3-37 2«-5« As it appears that 5 is the only common measure, the shortest method of finding the answer from the factors is, to divide the terms of the original fraction, ////o¥o , by 5. ^ns. ^UH- PROBLEM 3. To multiply two or more fractions togeth& and obtain their product in its lowest terms, by one operation. RULE. Arrange the prime numerators and the prime factors of the com- posite numerators on a line as the factors of the numerator of a single fraction; and the prime denominators and the prime factors of the composite denominators on a line as the factors of the denominators of the same fraction — then proceed by the third rule of Problem 3. THE USE OP PRIME FACTORS. 61 EXAMPLES. 1. What is the product in its lowest terms of the fractions ^, f , f and I? 2-3-2-^ _ 1 2-3-2=-5 5' 2. What is f of I ? ?:i — J_ 2^-3 2 ■ 3. Whatisf.of Tt.of if? 5-3^-2-3' _ 2-3^-5 __ l-3'-5 _ 45 2'-2*-2=-3»~ 2»-3= ~ 2»-l. ~256" 4. What is the product of {i, f i, iff and f f |i ? 17-3^- 3-5-7-2341 _ 3^-5-7-17-234 1 _ 17-2341 _ 39797 19^F3"^-5-7»-43- 109 "~ 3=-5-7=- 19-43- 109 ~" 3''-7- 19-43- 109 ~" 5610339' PROBLEM 4. To divide one fraction by another and obtain the quotient in its lowest terms, at one operation. RULE. Invert the terms of the divisor, and then proceed by the rule of Problem 3 for multiplication. EXAMPLES. 1. Divide ^ by xMir- 66_2-3-ll. ^^^ 1000 _y-y 100 2=-5' ' 83 ~3-ll" 2*-3-5»-ll 2'-5_ The factors of the quotient are oe.o.fis. TT ~^ ^ — 20 ^ns. For proof multiply jUs V 20. AW = AV . the dividend. 3. Divide 4| by l^. ., 14 2-7 , . ,_ 6 _2-3 4|=— =-g-; and 1^=1, mverted — y — -y 2°"3'7 The factors of the quotient are -5-=- = 4 jlns. Proof I X 4 = V = 4|, the dividend. 3. Divide || of a ship equally among 6 persons. =-^7 = ^:7, part to each. ^ns. 2 2'3 2 32 4. Divide IHf by f 5'-271-7 _' 271 _ 271 8-5»-7- 17-2-3 ~ 2-3»- 17 ~ 306 PROBLEM 5. To reduce reducible ratios of Proportion to their lowest terms with a view to abbreviate the work requisite for the solution of questions. DEFINITIONS AND EXPLANATIONS. The rules for reducing the ratios of Proportion do not differ in prin- 62 THE USE OP PRIME PACTORS. ciple, from those already given for reducing fractions and ratios in general. But, as questions to be solved by the Rules of Proportion may con- tain more than one reducible ratio, or a ratio compounded of several terms, it is proper to prescribe some uniform mode of expressing this ratio, so that its terms may be reduced by means of their prime fac- tors ; and so that the ratio, when expressed, whether reducible or not, may correctly indicate the value of the answer required, and also the work to be performed for obtaining it. Proportion arises from a comparison of the terms of equal ratios. From the explanations given at the beginning of this Chapter, it is easy to understand, that when two ratio's are equal, the relative mag- nitudes of their terms are also equal. As the relative magnitudes of each of the ratios is measurable by a fraction or portion of a unit, the equal ratios are, with propriety, called a proportion, that is, portion for portion. The four numbers, therefore, of any two equal ratios, properly arranged, do constitute a proportion. A number of diflferent proportions, indeed, may be form- ed from them. A perfect proportion consists of an even number of terms, four, six, eight or more. In every question, however, to be solved by the Rules of Propor- tion, one term of one of the ratios is not given, being that which is unknown or required to be found ; so that the number of given terms is odd, three, five, seven or more. The simplest form of a question of Proportion is that which con- tains but three given numbers to find a fourth. Hence the rule for solving questions of this form, long ago, acquir- ed the name of the Rule of Three, and, for its great utUity, also that of THE Golden Rule. It is, moreover, sometimes called The Single Rule of Three, or Simple Proportion ; while that for solving questions containing a greater number of terms, is called The Double Rule of Three, or Compound Proportion. Although there are, at the least, two diiferent ways of stating the ratios of the question with a view to its solution, yet the operations performed to obtain the answer must be essentially the same. That form of stating them will be given which the author deems the most rational and simple. In every question of Proportion one of tjie given numbers or terms is, in a concrete sense, like to that of the answer required. It may be call- ed homogeneous with the answer. To designate it by a name, it will be called, the answer's correlative term, or briefly, the answer's correlative. THE USE OP PRIME FACTORS. 63 This term and the answer will always constitute one of the ratios involved in the question, of how many soever terms it may consist. In any question of three terms only, the remaining terms, likewise homogeneous, will constitute the other ratio, which may be called the given ratio, of the proportion. The Problem of the Rule of Three, therefore, may be propounded in this form : Given the two terms of one ratio and one term of another equal ratio to find the other term of this ratio. The given ratio is the guide to the operations to be performed upon the answer's correlative term, in order to bring out the answer itself. In some questions this ratio is so simple, that as soon as announced, it suggests the answer ; while in others, some skiU and calculation are necessary to arrive at it. When either of the terms of the given ratio is 1, the answer may always be found by Multiplication or T)ivision only. Suppose the question to be : If a bushel of wheat cost one dollar, what will a thousand bushels ■cost 2 The answer, momentarily perceived, is one thousand dollars. The given ratio, formed from the relative quantities of wheat, being 1 : 1000, or '-"-j-"-" ; that for the corresponding amounts of money must be the same. The operation performed in this instance, upon the answer's correlative term, One dollar, though merely mental, is that of Multiplication. Suppose the question now to be : If a thousand bushels of wheat cost a thousand dollars, what is that per bushell The answer, palpable as before, is one dollar. The given ratio in this case, 1000 : 1 or x^Vo' i^ the inverse of that in the former, and the operation performed upon the answer's cor- relative term, 1000, is Division. But when both of the terms of the given ratio are greater than 1, the work, however it may be indicated or performed, must involve both Multiplication and Division ; that is, the multiplication of the answer's correlative by one of the terms of the given ratio, and the division of the product by the other ; or the division of the correlative by one term of the given ratio and the multiplication of the quotient by the other. The order of the operations is immaterial, except in regard to the facility or despatch of performing them. As it must always be evident which of the given terms of the question is the answer's correlative, whether it relate to a thing nat- ural or artificial, real or imaginary, the only other point to be ascer- 64 THE USE OF PRIME FACTORS. tained before commencing the work of solving the question, or of in- dicating by a proper collocation of the term's, how it is to be done, is to determine which of the terms of the given ratio is to be made the multiplier and which the divisor. This it can never be difficult to do ; for if the answer must be greater than its correlative, the greater of the terms of the given ratio must be made the multiplier, and the less, the divisor; and vice versa, if the answer must be less than its correlative, the less of the terms of the given ratio must be the multiplier, and the greater the divisor. To illustrate the truth of this position, a question will now be pro- posed for solution : Jf5 barrels of flour cost '27 dollars, how much will 630 barrels costl In this question, 27 dollars is the answer's correlative. As 630 barrels of flour must come to more than 5 barrels, it is appa- rent that 27 must be multiplied by 630, and the product, or one of its factors, be divided by 5, to obtain the answer. To express the value of the answer and the processes requisite for finding it, the numbers are set down in the form of a fraction or ratio thus : — — This ratio expresses the relation of a dividend to its 5 divisor, or the value of a quotient, which when found, will be the answer. To abbreviate the work the ratio may be first reduced to its lowest terms. It being apparent that 630 is the multiple of 5, the reduction may be made without the aid of the tables, by dividing these numbers by 5, their greatest common measure and retaining their quotients. The reduction being made, the ratio becomes ^ ^^ =: 27* 126 == 3^-126 =: 3402, the answer sought. Suppose now the question to be : If 630 barrels of flour cost 3402 dollars, what would five barrels cost ? The answer's correlative, now, is 3402. As it is evident that 5 barrels of flour cost less than that sum, which is the price of 630 barrels, the multiplier must be 5, and the divisor 630. The ratio, or fraction expressive of the value of the answer, there- fore, is ^^"^X5 ^jjjj.jj jg reducible, at sight, to ?^. bdi)^ 126 To reduce this last ratio to its lowest terms, the prime factors of its terms are found in the table of factors of even numbers and substitu- ted for the terms themselves, thus — - = 3' = 27, the answer 2-3=-7 sought. The smaller the quotient the more the ratio may be reduced in some cases. THE USE OP PRIME FACTORS. 65 Having fully shewn the method of arranging the terms of a ques- tion in Simple Proportion, we shall proceed to the consideration of questions of Compound Proportion. If 5 barrels of flour furnish, bread for 6 men 8 months, how many barrels will suffice for 144 men 12 months? The answer's correlative in this question is 5. Of the four remaining numbers two ratios are to be formed, each of terms homogeneous ; that is, one from the numbers relating to men, the other from the numbers relating to time. The effect which each of these ratios ought to have upon the answer's correlative may be separately considered. Each, it is evident, must increase the quantity of flour to be consum.ed. The ratio expressive of the answer, therefore, is — 5 X 144X12 _ 5X2-3^ ><_2-3 ^ 5;2^:3^ _ , , _ 6X 8 2-3 X2' 2^-3 — ^ ^ — 1»IJ -^ns. The last question is resolvable into two questions in Simple Pro- portion, which may be propounded and solved as follows: ^ If 5 barrels of flour furnish bread for 6 men 8 months, how many barrels will suffice for 144 men for the same time ? The time is an immaterial term in the solution of this question ; the ratio of the times being that of equality. The answer's correlative is 5, and the terms of the given ratio are 5 V 144 6 : 144. The answer is ^^^ = 120. The second question will now be : If 144 men consume 120 barrels of flour in 8 months, how many barrels would they consume in 12 months'! In this question the term 144 is immaterial to the answer, the numbers relating to men being equal. 120 X 12 The ratio formed of the other terms is g ^ 180, the answer ; which is the same as that to the original compound question. 5 V 144 In the solution of the compound question, the fraction g was used, instead of its equivalent number, 120, which rendered it unne- cessary to obtain this number before the ratio in regard to time was used. It may be seen that the value of the factors of the ratio form- ed from all the terms of the compound question is the same as that of the ratio formed of the three terms of the second simple question. It is proposed now to consider a similar compound question with one more given ratio. If 6 men in 8 months consume 5 sacks of flour weighing 150 pounds each, how many sacks of 175 pounds each, will suffice for 144 men 12 months. 9 66 THE USE OF PRIME FACTORS. This question has seven terms, the answer's correlative 5, and three given ratios; one ol men, one of sacks and the other of the weights ot the sacks. The effect of each ratio upon the number of sacks is to be consider- ed. The statement of the terms is as it was in the former compound question, except as to that relative to the weights of the sacks. It will not require so many of the larger sacks as it would of the smaller ones for any equal numbers of men for any equal times. The less one of the numbers relative to weights, therefore, must be set down in the numerator, as a factor of the dividend, and the greater one, in the denominator, as a factor of the divisor, of the compound ratio .,,,,, ,, 5X 144X12X150 equivalent to the answer, thus : 6 V 8 V 175 ' To simplify the work of reduction in this case, the given ratios may be first separately reduced, so far as they can be at sight, with- out resorting to the tables of prime factors. Reductions of the given ratios being separately made, the resulting equivalent ratios are — 5X24X 3X6 1080 _^, ^ 1^-2^ = -7-=^^^^-^~- From the explanations given it will be easy to state the terms of any question in Compound Proportion, however complicated it may be in appearance, by the aid of the following directions. Rules for stating the terms and their factors. I. Set the answer's correlative term at the extreme left of all the terms of the numerator of the ratio or fraction, which is to be made equivalent to the answer. II. Of the homogeneous terms of each of the given ratios, set one as a factor of the numerator and the other as a factor of the denominator of the same ratio or fraction ; observing that if the effect of a given ratio ia to make the answer greater than its correlative, the greater of the terms is to be set in the numerator and the less in the denominator, and vice versa. III. Reduce the ratio formed to its lowest terms by the Rules oi Problem 2 of this Chapter. EXAMPLES. 1. If 15 masons working 9 hours a day, build a wall 30 rods long in 75 days, in how many days will 35 masons, working 15 hours a day, build a wall 100 rods long? In this example the given ratios being mentally reduced, before their terms are set down, and the prime factors of the resulting terms substituted for the terms themselves, the first ratio formed is very THE USE OF PRIME FACTORS. 67 simple, and is easily reduced to its lowest terms, by cancelling its common factors, without the use of the tables, thus : 75X3X3X2-5 450 „.„ , Am. ^,^5^3 = ^ = 641 days. 2.' If 15 masons, working 9 hours a day, build a wall 3 feet thick, 8 feet high, and 30 rods long; in how many days will 35 masons, working 15 hours a day, build a wall 5 feet thick, 10 feet high, and 100 rods long.? This example contains two more ratios than the preceding example does ; one in regard to the thickness and the other to the height of the respective walls. The effect of each of these ratios upon the answer's correlative, time, must be considered. The reduced terms of the other ratios or their ascertained value ii-a, may be assumed from the work of the preceding example. 450X5X2-5 _ 2'3''-5''X5X2-5 _ 2^'3^-5'' _3-5<_1875_ Jlns. 7^3-^23 — 7X3X2" — 2'-3-7 — 2-7 — 14 — l^^T?- These examples are sufficient to shew how other questions of Pro- portion, simple or compound, involving larger numbers, may be solv- ed by the aid of the tables or otherwise. CHAPTER V. OF ROOTS RATIONAL AND IRRATIONAL OR SURD. DEFINITIONS AND EXPLANATIONS. Every numerical root which can be exactly expressed, is called rational ; but every one which cannot be so expressed is called irra- tional or surd. All the rational roots of whole numbers are whole numbers. For every power of a fraction or mixed number, is necessarily also a fraction or mixed number. Prime numbers have no rational roots of any degree. Comparatively few of the composite numbers have rational roots. For 100« = 10,000 and 100' = 1,000,000. And SOO'' = 40,000 " 200' = 8,000,000. Whence it appears that the number of rational square roots of whole numbers, from 1 to 10,000 inclusive, is only 100; and of numbers from 10,000 to 40,000, it is the same. 68 THE USE OF PRIME FACTORS. And that the number of rational cube roots of whole numbers from 1 to 1,000,000 inclusive, is 100; and of numbers from 1,000,000 to 8,000,000, it is the same. Of integral numbers below 20,000, the greatest rational square root is 141, whose square or second power is 19881 ; the greatest rational cube root is 27, whose cube or third power is 19683; the greatest fourth root is 11, whose fourth power is 14641. The roots of a lower degree are very few. It is easy to determine which of the composite numbers in the tables have rational roots of any degree from the indices of their prime factors. For the index of the square or second power of any number, prime or composite, is double that of the root; of the cube or third power three times that of the root; of the fourth, four times, &c.; and the indices of the prime factors, whether one or more, of any powers of any degree must have a corresponding relation to those of the roots. Wherefore no composite number has a rational square or second root unless the index of every one of its prime factors is either 2, or . some multiple of 2, as 4, 6, 8, &c., that is, in short, unless it is an even number. No composite number has a rational cube or third root unless the index of every one of its prime factors is 3 or some multiple of 3, as 6, 9, 12, &c. The law is similar in regard to any other roots. No composite number, therefore, has a rational root of any particu- lar degree, unless the index of every one of its prime factors, as de- signated in the tables, is divisible by the number indicative of the root, without a remainder. In order that any composite number may be a perfect power of any two or more roots of different degrees, the index of each one of its prime factors must be a common multiple of the numbers indicative of such roots. If the index of each of the prime factors of any number is 6, or any other multiple of 2 and 3, the number has a rational square root and also a rational cube root. And if the index be 12, or any other multiple -of 2, 3 and 4, the number has a rational square, cube, and fourth root. THE USE OP PRIME FACTORS. 69 PROBLEM 1. To determine whether a given number has or has not a rational or exact root of a particular degree. RULES OR CRITERIA. I. If the given number be a prime number, it has no rational root of any degree. II. Though the given number be a composite number, it has no rational root of a particular degree, unless the index of its prime fac- tor, or of each of its prime factors, is divisible by the number indica- tive of the root of that degree, without a remainder. EXAMPLES, 1. Has the number 941 any rational square or cube root ? Ans. No — because it is a prime number. 2. Has the number 19321 a rational square root? Ans. Yes. Its prime factor is 139*. 3. Has the number 19881 any rational square root.? Ans. Yes. Its prime factors are 3^-47^. 4. Has the number 7744 any rational square root .? Ans. Yes. Its prime factors are 2^'IP. The indices are both divisible by 2 without a remainder. ' 5. Has the number 19683 any rational cube root ? Ans. Yes. Its factors are 3'. Its cube root 3'. 6. Has the number 17576 any rational cube root ? Ans, Yes. Its prime factors are 2'" 13'. 7. Has the number 8784 a rational square root .' Ans. No. Its prime factors, 2^'3''-61 ; the index of only two of them is divisible by 2 without a remainder. 8. Has the number 8712 any rational cube root .? Ans. No. Its prime factors are 2^* 3'''ir. The index of only one of them is divisible by 3. PROBLEM 2. To find any rational root of a composite number that has' such root. RULES. I. Divide the index of each of the prime factors of the given number by the number indicative of the root required, and reserve the quotients. II. In lieu of the indices of the prime factors of the given num- ber, substitute for indices the respective quotients found, and the same factors, with these indices, wiU be the factors of the root required. 70 THE USE OF PRIME FACTORS. III. From these factors and their indices, find their equivalent number, prime or composite, and it will be the root required. EXAMPLES. 1. What is the square root of 19321 ? Its prime factors are 139'. •dns. 139. 2. What is the square root of 19881 ? Its factors are 3»-47'- Jlns, 3-47= 141. 3. What is the square root of 7744 ? Its factors are 2°-lF. , Jlns. 2'-ll = 88. 4. What is the cube root of 19683 ? Its factors are 3«. Jlns. 3' = 27. 5. What is the cube root of 17576 ? Its factors are 2'- 13'. ^ns. 2-13 = 26. 6. What is the cube root of 5832 ? Its factors are 2'-3«. Jlns. 2-3"= 18. 7. What is the fourth root of 10,000 .' .3ns. 2-5 = 10. 8. What is the fifth root of 1024 > Jlns. 2' =; 4. OF THE REDUCTION OP SURDS. Although surds are roots that cannot be exactly expressed or ob- tained, yet some of them are resolvable into two factors, of which one is rational while the other is still irrational or surd. As for in- stance, ^45 = ^9 X \/5 = W^ ; and 4/48 = ^16 X \/3 = 4^3- These forms are called the simplest terms of the original surds. The Rules and Examples of the two following Problems are de- signed to shew how readily surds which are reducible may be distin- guished from those which are irreducible ; and how those which are reducible may be reduced to their simplest terms by means of the indices of the prime factors of their respective powers, or rather by means of the composite measures of the powers indicated by these indices. PROBLEM 3. To distinguish reducible Jrom irreducible surds. RULES OR CRITERIA. I, When the power is a prime number, the root is not reducible to simpler terms. II. When the power is a composite number, which contains among its measures a perfect power of the root, the whole root or surd is reducible to a simpler form, otherwise not. THE USE OF PRIME FACTORS. 71 EXAMPLES. 1. Is y'SST reducible to simpler terms ? Ans. No. The power is a prime number. 2. Is s/^m reducible ? Ans. No. The factors of the power are 3-7-17, neither of which is a square number, or a poWer containing a square number. 3. Is v'SSa reducible? Ans. Yes. Its factors are 3^-37; one of which contains a square number. For 3' = 3-3'; 4. Is v'4563 reducible ? Ans. Yes. For 4563 = 3='-13» = 3-3'-13^ two of these measures are square numbers whose roots are 3-13 = 39. 5. ' Is ^8928 reducible ? Ans. Yes. The factors of the power are 2'-3'-31 = 2-2'-3''-31. Two of these measures, 2'''3', are perfect square numbers, and their roots 2=-3=:12. 6. Is ^9900 reducible? Ans. Yes. 9900 = 2=-3^-5^-ll. 7. Is ^9920 reducible? Ans. Yes. The factors of the power are 2°-5-31, and 2° is a perfect cube or third power of 2'' = 4. 8. Is ^9000 reducible ? Ans. Yes. 9000 =z 2'-3''-5' ; and the cube root of 2'-5' = 2-5 = 10. 9. Is ;^9984 reducible? Ans. Yes. 9984 — 2«-3- 13, and the fourth root of 2* =z 2" = 4. 10. Is any root of 7990 = 2-5- 17-47, reducible ? Ans. No. Neither of its measures is a perfect power of any degree. PROBLEM 4. To reduce reducible surds to their simplest terms. RULES. I. Find, by the rules of Problem 2, the rational root of such factor or factors of the given power as have such root, and put the other factor or the product of the other factors under its proper radical sign or index. 11. Substitute the rational root as one of the factors of the surd, in lieu of its equivalent factor, under the radical sign or index, and the rational factor and the remaining surd factor will constitute the simplest form of the original surd. EXA,MPLES. 1. Reduce v'4563 to its simplest terms. The power 4563 = 3'- 13^ = 3- 3^- 13^ And \/3-"3M3^=V3X\/3'' X^/13^ Andx/3^X\/13'' = 3-13=i:39. ^w*. 39^3. 72 THE USE OF PRIME FACTORS. 2. , Reduce \/6975 to its simplest terms. 6975 = 3=-5^-31. And ^Z'Xs/S' =z 3-5 =: 15. Jlns. IS^/Sl. 3. Reduce ^^8928 to its simplest terms. 8928 = 2'-3'-31 = 2-2^-3=-31. And ^§928 = ^/2^X^/3»X^/2X V31 = 12^62. 4. Reduce ^^9900 to its simplest terms. 9900 = 2'-3''-5«-ll, and v/9900=V2'Xv'3'X\/5'X\/ll = 30Vll. 5. Reduce ^45, ^125 and ^405, to their simplest terms. 45=3«-5 and ^/45 = 3^5; 125 = 5=-5 and V125 = 5v'5; 405= 3*-5 and V405 = v^3^Xv'5 = 9v'5. 6. Reduce v' 19845 to its lowest terms. 19845 = 3^-5-7'' and x/19845 = V3^Xv'7''X\/5 = 63V5. 7. Reduce ^9000 to its lowest terms. 9000 = 2'-3''-5' and v'9000 = ^2'X^5'XV3'= 10/^/9. 8. Reduce ^^19551 to its simplest terms. 19551 = 3-7^-19 and 11 13 13 13 17 17 21 17 19 ""23 19 * "23 19 21 19 19 21 23 23 23 " "27 27 29 29 29 31 29 31 31 " "37 33 33 33 33 37 37 39 41 39 41 43 47 43 43 47 " "49 47 47 49 ■_ 51 >53 57 59 51 53 ""59 61 57 61 63 67 69 57 59 59 61 " " '67 61 • 63 67 69 63 " "67 67 69 71 71 73 71 71 73 77 " "79 77 79 ' " "83 " "89 81 81 83 81 87 89 91 89 89 91 > 91 " "93 91 " "97 93 97 99 99 No. 12 10 11 15 IJ 14 13 12 11 11 78 PRIME NUMBERS. From 4000 to 4100 4100 4200 4200 4300 4300 4400 4400 4500 4500 4600 4600 4700 4700 4800 4800 4900 4900 5000 s 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 S3 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 3 7 1 1 3 3 3 7 9 9 "'"13 11 11 13 17 19 '""23 13 17 ""19 17 19 19 21 21 23 21 21 23 27 27 29 "'"29 31 27 29 31 31 33 37 33 33 37 39 37 39 39 41 43 41 , 43 43 47 51 47 49 49 51 " " "57 49 49 51 51 51 53 57 59 53 ' " "59 61 57 57 ■ ■••■■ 57 67 59 • ■•••> 61 ""67 61 ■■«■•• 63 63 63 67 69 " "73 71 73 71 '"73 " "79 '""77 73 73 77 79 , 81 83 83 83 83 87 89 " "89 87 89 91 93 91 " "93 91 91 93 93 97 97 97 99 99 99 No. 15 9 16 9 11 12 12 12 8 15 — ' PRIME NUMBERS. 79 From 5000 to 5100 5100 5200 5200 5300 5300 5400 5400 5500 5500 5600 5600 5700 5700 5800 5800 5900 5900 6000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 No. 3 9 11 1 1 3 7 1 1 3 3 7 7 7 9 9 11 13 13 17 19 13 17 21 23 19 19 21 21 ' " ' "23 27 23 23 27 27 27 . , 31 33 37 " "33 31 31 37 37 41 43 39 39 41 .39 39 41 43 43 47 47 47 49 49 49 51 "53 51 51 51 53 57 59 53 57 57 59 61 61 63 67 67 69 69 69 71 71 73 73 77 81 77 79 " "83 79 79 81 81 79 79 81 81 81 83 83 87 87 87 89 89 " "93 91 91 93 " ■ '99 97 97 97 99 12 11 10 10 13 13 12 10 16 7 80 PRIME NUMBERS. From 6000 to 6100 6100 6200 6200 6300 6300 6400 6400 6500 6500 6600 6600 6700 6700 6800 6800 6900 6900 7000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 S3 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 3 1 1 3 3 "••j 7 7 9 11 "ii 11 11 11 17 17 17 19 19 21 21 " "23 21 21 23 27 29 27 29 " " '37 31 33 29 29 29 33 37 33 37 37 41 43 47 43 " "47 43 47 47 49 49 51 ' * "53 51 51 53 53 53 57 57 "59 61 "*67 71 59 61 59 61 61 63 " "63 ""67 63 63 63 67 69 71 ""77 69 69 71 77 69 71 73 73 73 73 73 77 79 79 79 79 81 81 81 83 83 i 87 89 91 89 89 91 91 91 93 91 97 99 ' "99 97 97 99 99 No. 12 11 13 15 8 11 10 12 12 13 PRIME NUMBERS. 81 From 7000 to 7100 7100 7200 7200 7300 7300 7400 7400 7500 7500 7600 7600 7700 7700 7800 7800 7900 7900 8000 1 '■ 1 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 3 3 7 3 7 11 13 7 9 7 7 9 11 13 17 17 17 17 ""19 19 91 19 21 21 23 23 27 23 ' ■ "99 ""hi 27 27 29 29 29 31 33 33 33 37 37 37 39 39 ■ " " '43 41 41 41 43 43 47 47 49 49 51 51 49 49 51 51 ""53* 53 57 59 53 57 57 59 59 59 61 63 67 69 69 69 73 77 73 73 77 79 ' " "79 77 77 81 " " "83 " "89 91 81 83 83 87 87 89 87 91 89 93 ' " "97 93 93 93 99 99 No. 9 10 11 9 11 15 12 10 10 10 11 82 PRIME NUMBERS. From 8000 to 8100 8100 8300 8200 8300 8300 8400 8400 8500 8500 8600 8600 8700 8700 8800 8800 8900 8900 9000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 1 3 7 7 9 11 11 9 9 11 13 13 17 V 17 "'"'ig 21 17 19 19 19 21 ""23 21 " ' " '27 33 23 23 27 29 29 29 31 29 """'33 41 31 33 37 31 31 37 39 37 37 39 39 1 41 41 43 43 47 43 47 47 47 49 51 53 63 53 59 61 61 ' " " 67 61 61 63 67 ""'63 '""69 71 63 63 63 63 " "69 67 71 69 69 69 73 73 77 77 81 79 79 81 81 83 87 89 "m 87 87 89 87 89 91 91 93 97 93 93 97 99 99 99 1 No. 11 10 14 9| 8 12 13 11 13 9 PRIME NUMBERS. From 9000 to 9100 9100 9200 9200 9300 9300 9400 9400 9500 9500 9600 9600 9700 9700 9800 9800 9900 9,900 10,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 No. 1 7 11 13 1 1 7 3 3 3 3 9 9 11 ""ii 11 11 13 17 19 " " 23 19 21 21 19 ""23 19 21 21 23 ' " " "29 27 27 29 31 29 29 31 31 33 37 39 ""33 33 37 33 33 " " '39 41 37 41 43 39 39 39 41 41 43 43 43 47 49 49 49 49 51 49 51 51 " "59 57 57 57 59 61 61 63 67 61 67 67 69 67 71 71 ""73 73 73 77 77 77 79 79 81 81 83 81 83 87 ...... 87 87 " ■ '89 87 91 91 91 91 93 97 97 97 99 11 12 11 11 15 7 13 11 12 9 84 PRIME NUMBERS. From 10,000 to 10,100 10,100 10,300 10,200 10,300 10,300 10,400 10,400 10,500 10,500 10,600 10,600 10,700 10,700 10,800 10,800 10,900 10,900 11,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 3 1 1 7 9 3 3 7 9 11 9 11 11 13 13 13 21 23 23 27 29 ""33 " "29 31 27 31 29 ""33 31 33 37 31 " "37 39 33 37 37 39 39 41 39 39 43 47 43 47 ""49 51 51 53 " "57 53 57 59 53 53 " "57 " "59 57 61 ""67 69 59 59 59 61 63 " "69 63 ""67 63 67 67 71 73 67 69 71 73 ""79 1 77 77 79 81 81 83 ""87 ""93 87 " "89 87 91 89 89 89 91 91 93 91 93 97 99 99 99 99 No. 11 12 10 12 10 8 12 11 10 10 PRIME NUMBERS. 85 From 11,00C to 11,10C 11,10C 11,20C ll,20f 11,30C 11,30C 11,40C 11,400 11,500 11,500 11,600 ]1,60C 11,700 ll,70C 11,800 11,800 11,900 11,900 12,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 • 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 1 3 9 3 7 11 11 13 17 19 13 13 17 17 21 17 19 19 21 21 " "23 27 23 27 27 27 29 31 31 31 33 " " "33 33 37 39 39 39 41 43 43 47 43 47 49 49 51 51 51 53 53 57 59 " " '59 61 57 57 59 61 63 67 ""69 71 67 69 71 69 71 73 77 71 73 77 81 77 79 79 79 81 83 87 83 83 83 87 87 87 87 89 91 """97 89 89 93 93 93 97 97 97 99 99 99 No. 10 11 10 10 11 9 8 9 12 13 86 PRIME NUMBERS. From 12,000 to 12,100 12,100 12,200 12,200 12,300 12,300 12,400 12,40C 12,500 12,50C 12,60C 12,60C 12,70C 12,70C 12,80C 12,80( 12,90C 12,900 13,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 No. 1 3 1 1 3 1 3 7 11 7 ■■"'is 7 11 9 9 11 11 ""17 11 13 13 13 17 19 •■-23 19 19 21 21 21 23 23 27 27 29 29 33 37 37 ■'"39 41 37 41 39 41 39 ■'"43 41 43 ' " "43 41 41 i 43 47 47 47 49 49 51 53 51 53 53 ■ ' " '57 53 53 57 57 59 59 61 63 63 63 67 69 69 71 73 71 73 77 79 73 ■""79 73 ' 77 77 79 81 81 83 83 87 89 89 89 91 89 91 91 93 97 97 97 97 99 99 9 11 12 9 13 12 12I 10 9 12 PRIME NUMBERS. 87 From 13,000 to 13,100 13,100 13,200 13,200 13,300 13,300 13,400 13,400 13,500 13,500 13,600 13,600 13,700 13,700 13,800 13,800 13,900 13,900 14,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 No. 1 3 7 9 1 3 7 3 7 9 9 9 11 11 ""17 13 13 13 13 17 19 19 21 21 21 23 21 23 " "27 27 29 27 29 29 31 31 33 31 33 37 33 37 39 37 41 41 41 43 " " "49 47 51 49 49 51 51 53 57 57 59 59 "59 59 63 63 63 ""69 63 .^ • . ■ . 63 67 67 _ 67 67 69 71 73 77 79 77 77 77 79 81 81 81 83 87 83 87 87 89 91 . 91 91 93 97 93 97 97 99 " "99 97 97 99 99 99 11 12 ,9 10 11 8 12 12 ( " 88 PRIME NUMBERS. From 14,000 to 14,100 14,100 14,200 14,200 14,300 14,300 14,400 14,400 14,500 14,500 14,600 14,600 14,700 14,700 14,800 14,800 14,900 14t90D' 15,000= 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 7 3 3 9 11 7 7 11 13 17 13 19 19 21 21 23 27 31 '""23 21 27 ""23 ""29 23 27 29 29 31 31 31 33 33 37 33 " "39 37 37 39 41 41 43 43 43 43 ""47 47 47 49 47 51 ""si 49 " "53 49 51 49 51 51 51 53 57 53 57 57 59 59 61 61 63 67 71 67 69 ""69 69 69 71 73 77 79 79 79 81 83 87 81 83 83 ""87 83 87 89 89 91 93 91 93 97 97 97 99 No. 10 8 7 9 12 12 10 14 12 8 PRIME NUMBERS. Prom 15,000 to 15,100 15,100 15,2Q0 15,200 15,300 15,300 15,400 15,400 15,500 15,500 15,600 15,600 15,700 15,700 15,800 15,800 15,900 15,900 16,000 1 3 7 9 11 13. 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 1 1 1 7 3 7 7 7 a 11 13 17 13 13 13 ""'19 17 17 19 19 21 23 23 27 29 31 27 27 ■'"29 27 31 31 '""33 31 33 37 39 37 39 37 39 41 41 41 43 47 49 43 49 49 49 51 51 53 59 '""63 59 61 61 59 59 59 61 61 61 61 67 '""69 67 67 • 69 71 ""77 f 71 71 73 73 77 73 73 77 73 73 ' " "77 79 81 83 81 83 83 83 87 87 89 87 87 89 91 91 91 91 93 93 97 97 99 99 No. 9 12 12 12 11 8 13 12 9 ]0 12 90 PRIME NUMBERS. From 16,000 to 16,100 16,100 16,200 16,200 16,300 16,300 16,400 16,400 16,500 16,500 16,600 16,600 16,700 16,700 16,800 16,800 16,900 16,900 17,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 7 1 1 3 3 3 7 3 11 11 11 17 """19 17 21 19 19 21 ""27 31 23 23 ■ -••■•• 27 27 29 31 29 31 33 29 29 31 33 33 33 37 39 41 30 41 43 43 47 47 ""49 51 47 49 49 51 53 ""53 53 57 57 59 61 63 67 69 61 63 61 61 63 63 67 67 69 71 73 73 73 73 77 79 79 81 ■■■'87 81 81 ' " "87 91 " "97 83 87 89 83 ■■"89 87 87 91 93 93 93 93 99 N.. 12 9 8 9 13 7 13 7 9 12 PRIME NUMBERS. 91 From 17,000 to 17,100 17,100 17,200 17,200 17,300 17,300 17,400 17,400 17,500 17,500 17,600 17,600 17,700 17,70C 17,800 17,800 17,900 17,900 18,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 SI 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 3 7 9 3 9 11 7 7 7 9 9 11 13 17 17 17 19 19 21 ' " "27 29 21 ' 21 23 ' " " '29 23 28 27 27 ' " "29 27 4 31 ""33 31 33 S7 37 37 39 "■'39 39 39 41 41 43 47 47 49 49 51 51 51 53 57 57 59 57 59 59 59 61 63 67 67 69 69 71 71 73 77 77 77 77 79 81 81 83 ■""83 81 81 83 83 87 89 ""93 83 87 89 " "93 89 91 91 93 89 91 89 91 91 97 97 99 99 99 No. 11 9 9 12 13 10 8 10 8 14 92 PRIME NUMBERS. Prom 18,000 to 18,100 18,100 18,200 18,200 18,300 18,300 18,400 18,400 18,500 18,500 18,600 18,600 18,700 18,700 18,800 18,800 18,900 18,900 .19,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 1 1 3 3 7 ....•• 11 11 13 U 13 17 19 13 13 13 17 n 17 • • a^i* • 19 21 19 21 23 23 27 27 ...... 29 29 31 33 31 33 33 37 39 ""43 39 41 39 41 43 47 49 41 ■■•>•• 43 43 47 • 49 49 51 53 57 """53 51 ""57 ^ 53 57 59 61 59 59 61 61 67 69 69 69 71 71 73 73 77 79 79 79 81 81 83 87 87 89 87 89 91 91 93 93 93 97 97 97 99 99 No. 10 111 11 11 11 10 6 11 5 8 PRIME NUMBERS. 93 Prom 19,000 to 19,100 19,100 19,200 19,200 19,300 19,300 19,400 19,400 19,500 19,500 19,600 19,600 19,700 19,700 19,800 19,800 19,900 19,900 20,000 1 3 7 9 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 1 1 3 1 ...... 1 3 7 9 9 9 9 ****** 11 13 , 13 •13 13 17 17 19 19 19 19 21 21 23 27 29 27 27 31 31 31 33 33 37 " "39 41 37 37 39 41 41 43 41 43 47 49 ...... 49 51 51 53 53 53 57 57 59 59 61 59 ""63 61 ""67 61 63 63 63 67 69 69 71 71 73 73 73 73 77 77 77 79 81 ■"*87 79 81 79 81 83 81 83 83 87 91 87 89 89 89 91 93 97 93 97 97 99 No. 11 7| 11 9 16 11 7 10 10 12 TABLE OF THE PRIME FACTORS COMPOSITE ODD NUMBERS FROM 1 TO 20,000. 96 PRIME FACTORS From to 100 100 200 200 300 300 400 400 600 1 3 5 7 9 11 13 15 17 19 21 23 2S 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-67 7-29 5-41 3«-23 11-19 7-43 3-101 5-61 13-31 3<-5 11-37 3-5-7 3« 3-103 3-37 3-137 7-59 5-83 3-139 ■B 3-71 5-43 7-31 3-73 13-17 3-5 5-23 3' -13 7-17 IP 3-41 5' 3''-5-7 ii-29 3-107 17-19 5«-13 3-109 7-47 3-7 38-47 58-17 • 7-61 3-11-13 5' 3» 3'-5« 3-43 3-7-11 3-11 5-7 7-19 3^-5 38-37' 5-67 5-47 3-79 3-5-29 19-23 3-13 3-113 11-31 V 3-5-23 3-47 11-13 5-29 3-7'' 38-78 5-89 3-149 3' 5-7» 13-19 3-83 3^-5 3-17 3'- 13 11-41 3-151 5«7-43 3«-17 5-31 .11-23 3-5-'l'i' 5-11 3-19 5-71 3-7-17 3-53 7-23 7-37 3»-29 3^-17 19* 3- IP 5-73 3^-7 5-13 3-5-11 5-53 3-89 3-5-31 3-23 13^ 3' -19 38-41 T-53 7-67 3-157 11-43 58-19 38-53 3-7-13 5=^-11 3-58 7-11 5»-7 3-59 3-5' 13-29 3»-31 3* 3-127 5-7-ii 38-43 13.37 3-7-23 5-97 3-61 5-37 11-17 3^-7 3-5-19 7-41 17= 3-97 5-17 3-29 3.163 7-13 3-31 5-19 17-23 3-131 5-79 17-29 38-5-11 7-71 3-5-13 5-59 3'- 11 13-23 3='- 11 3-7-19 OF ODD NUMBERS. 97 From 500 to 600 600 700 700 800 800 900 900 1000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-167 .5-101 3-13^ 1-13 33-19 5-103 11-47 3-173 3-5^-7 17-31 23^ 3= -59 13-41 5-107 3-179 7^-11 181 109 3^-61 19-29 7-79 3-5-37 13-43 3-11-17 s'-iis ' 3<-7 3-191 5^-23 3-193 7-83 11-53 32-5-13 19-31 3-197 5-7-17 3-199 3^-67 5-11^ 3-7-29 13-47 3-5-4l' 3' -23 7-89 5* 3-11-19 17-37 3-211 5-127 7= -13 3^-71 3-5-43 11-59" 3-7-31 5-131 3^-73 3-13-17 5-7-19 23-29 3-223 11-61 33-5^ 7-97'" 3-227 5-137 3-229 13-53 32-7-11 5-139 17-41 3-233 19-37 3-5-47 7-101 32-79 23-31 5-11-13 3-239 7-103 3-241 52-29 3» 17-43 3.5-72 11-67 3-13-19 5-149 32-83 7-107 3-251 5-151 s'-ii-ss' 7-109 32-5-17 13-59 3-257 52-31 3-7-37 19-41 11-71 33-29 5-157 3-263 7-113 13-61 3-5-53 17-47 32-89 11-73 5-7-23 3-269 3-271 5-163 19-43 32-7-13 3-52-11 3-277 72-17 5-167 33-31 292 3-281 5-132 7-112 3-283 23-37 32-5-19 3-7-41 5-173 3-172 11-79 13-67 32-97 53-7 3-293 3-5-59 7-127 3*-ll 19-47 5-179 3-13-23 29-31 17-53 3-7-43 5-181 32-101 11-83 3-5-61 7-131 3-307 13-71 52-37 32-103 72-19 3-311 5-U-17 3 -sis'" 23-41 33-5-7 '13-73" 3-317 5-191 3-11-29 7-137 312 32-107 5-193 3-17-19" 7-139 3-52-13 11-89 32-109 5-197 3-7-47 23-43 3-331 5-199 33-37 13 PRIME FACTORS From 1,000 to 1,100 1,100 1,200 1,200 1,300 1,300 1,400 1,400 1,600 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63- 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 93 7-11-13 17-59 • 3-5-67 19-53 3-337 .5-7-"29' 3^-113 3-11-31 5'- 41 13-79 3-73 3'i-5-23 17-61 3-347 7-149 5-11-19 3-349 3*- 13 5-211 7-151 3-353 3-5-71 11-97 32-7.17 99-37 52-43 3-359 13-83 23-47 3-19^ 5-7-31 3=-lP 3-5-73 7- 157" 3-367 5-13-17 33-41 11-101 3-7-53 5-223 3-373 19-59 32-53 72.23 3-13-29 11-103 5-227 3-379 17-67 7-163 32-127 5-229 31-37 3-383 3-5-7-11 13-89 19-61 3'-43 5-233 3-389 7-167 3-17-23 52-47 11-107 32-131 7-132 3-5-79 29-41 3.397 5-239 32-7-19 11-109 3-401 5-241 17-71 3-13-31 7-173 3=-5 23-53*"' 3-11-37 52-72 3-409 32.137 5.13.19 3.7.59 17-73 11-113 3-5-83 29-43 32-139 7-179 5-251 3-419 13-97 3.421 5-11-23 7-181 3'.47 31.41 19.67 3-52-17 3.7.6I 5.257 32.11.13 3.431 5.7.37 3-433 32-5-29 7.'ii.i7" 3.19.23 13-101 5-263 3-439 33.72 52-53 3-443 IP 31-43 3-5-89 7-191 13-103 32-149 17-79 5-269 3-449 19-71 7-193 3-11-41 5-271 23-59 32-151 29-47 3.5.7.13 372 3.457 5».ll 3*. 17 7-197 3-461 5.277 19.73 3.463 13.107 7.199 32.5.31 11-127 3-467 23-61 5-281 3-7-67 17-83 32-157 5-283 13-109 3-11-43 72-29 3-52.'i9" 33-53 5-7-41 3-479 11-131 3-13.37 5-172 32.*7.23" 3.5.97 31.47 3.487 7-11-19 5-293 ' 32-163 13-113 3-491 52-59 7-211 3-17-29 33-5-11 3.7.71 5-13-23 3-499 OF ODD NUMBERS. 99 From 1,500 to 1,600 1,600 1,700 1,700 1,800 1,800 . 1,900 1,900 2,000 ■ 1 3 , 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 19-79 3'- 167 5-7-43 11-137 3-503 1 3^-7 13-131 5-11-31 3-569 7-229 3-5-107 3-601 5-192 13-139 3^-67 72-37 3-5-112 23-79 17-107 3-607 11-173 3-5-127 23-83 3-72.13 3^-179 5- 17-" 19""" 3-72-11 29-59 3-571 5-73 17-101 32-191 17-89 3-5-101 37-41 72-31 3=-132 5-383 3^-71 ■19-101 17-113 3-641 52-7-U 41-47 3-643 3-541 53-13 5=-61 3-509 11-139 3-52-23 11-157 7-13-19 3-577 52-73 32-7-29 31-59 32-I8I 7-233 23-71 3-5-109 3-7-73 5-307 29-53 3" -19 23-67 3-13-47 5-367 11-167 3-613 7-263 19-97 32-5-41 432" 3-617 17-109 5-7-53 3-619 11-132 5-347 32-193 37-47 32-5-43 13-149 7-277 3-647 29-67 5-389 3-11-59 11-149 3-547 31-53 5-7-47 33- 61 17-97 13-127 3-19-29 5-331 3-7-83 5-349 3-il-'53 17-103 3-5-103 7-13-17 3-11-47 5-311 32-173 7-223 3-521 5-313 3-523 32-7-31 5-17-23 19-103 3-653 37-53 13-151 3-5-131 7-281 11-179 3' -73 3'-5-13 7-251 3-7-79 11-151 3-587 41-43 5-353 3-19-31 29-61 7-11-23 32-197 52-71 3<-23 5-373 32-5-37 3-7-89 3-557 7-239 5^-67 3-13-43 23-73 412 32-11-17 5-337 7-241 3-563 19-89 IP- 13 32-5'-7 19-83 3-5* 52-79 3-659 3-593 13-137 3-17-31 32-11-19 7-269 5-13-29 3-17-37 7-283 3-661 5-397 - 5-317 3-23= 7-227 37-43 3'.59 5-11-29 3-5-7-17- 32-13-17 11-181 32-199 11-163 5-359 3-599 7-257 31-61 3-631 5-379 7-271 32-211 3-5-113 3-5-7-19 3-13-41 100 PRIME FACTORS From 2,000 to 2,100 2,100 2,200 2,200 2,300 2,800 2,400 2,400 2,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-23-29 11-191 3-701 5-421 7' -43 3-19-37 31-71 3-13-59 72.47 5-461 3-769 74 3!'-89 5-13-37 29-83 3-11-73 5.401 3^-223 7= -41 3=-5-7« 47= 3-11-67 3-11-61 5-13-31 3»-257 5-463 7-331 3-773 11-211 23-101 3-55-31 13-179 17-137 3='-7-37 19-127 3-5-7-23 3=-5-47 29-73 13-163 3-7-101 11-193 55-17 3-709 5-443 3-739 7-317 3«-"l3-19 ' " " " 5«-89 17-131 3-743 23-97 7-11-29 3-5-149 3-673 43-47 7.172 3^-52 41-59 5=-97 3-809 7-347 11-13-17 3-811 5-487 3-677 19-107 5-11-37 3-7-97 13-157 32-227 5-409 23-89 3-683 7-293 3' -79 5-7-61 5-467 3-19-41 3-53-31 32-271 3^-83 3-11-71 5-7-67 7-349 3-5-163 3-5-11-13 19-113 7-307 3''-239 5-449 3-7-107 13-173 3"- 29 31-79 3-19-43 11-223 5-491 3'-7-13 3-751 5-11-41 37-61 3='-251 7-17-19 31-73 3-5-151 13-181 3-5-157 3-5-137 IP- 17 29-71 3= -229 5-431 3-719 17-127 h'-i'-m 5-433 11-197 3^-241 13-167 41-53 3-5^-29 7-311 7-337 3-787 17-139 5-11-43 3»-263 23-103 23-107 3-821 5-17-29 5-7-59 3-13-53 3-823 7-353 19-109 3-691 5^-83 31-67 3^-7-11 3-757 3-7-113 5^-19 5«-7-13 3=-ll-23 43-53 32-52-11 3-13-61 .37-67 3-827 13-191 5-7-71 3-829 19-131 47-53 32-277 5-499 11-227 3-72-17 3-727 37-59 5-19-23 3' 11-199 7-313 3-17-43 5-439 13' 3-733 3-761 5-457 3-5-139 3=^5-53 7-11-31 3.7.109 29-79 3-17-41 7-13-23 5-419 3^-233 3-797 3'-5-17 ii«-'i9 5-479 3-17-47 OF ODD NUMBERS. 101 From 2,500 to 2,600 2,600 2,700 2,700 2,800 2,800 2,900 2,900 3,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 41-61 3=- 17= 19-137 5-521 3-11-79 37-73 3-17-53 5-541 3-967 3-5-167 23-109 13-193 3*-31 7-359 5-503 3-839 11-229 3-5-11-17 7-401 53= 3-937 29-97 5-563 3= -313 5-7-83 3=-17-19 3=-7-43 7-373 3-13-67 5-523 41-71 3-971 5-11-53 3-5-181 11-13-19 3^-97 3-7-139 23-127 37-79 3=-5=-13 3 '907 7-389 5=-109 3'- 101 7-.13-31 3-941 5= -113 11-257 3-23-41 19-149 3^'-5-'7 17-167 3-947 3-29« 5^-101 7-19= 3=-281 43-61 3-53-7 37-71 11-239 3-877 29-101 3-977 7-419 5-587 3-11-89 17-149 3-5-13= 43-59 3-7-11= 3-911 5-547 7-17-23 3-11-83 5-17-31 3= -293 7-13-29 19-139 3-881 5-23= 17-173 33-109 5-19-31 7-421 3-983 13-227 13-211 3=-5-61 41-67 3-7-131 5-509 3= -283 5-569 3-13-73 7-11-37 3-883 11-241 7-379 3=-5-59 3-23-37 5-7-73 3=-317 5-571 5-19-29 3 -"919 31-89 11-251 3= -307 5-7-79 3-5-197 3-853 13-197 11-233 3'-5-19 17-151 7-367 3-857 31-83 5=-103 3-859 29-89 3=-7-41 5-11-47 13-199 3-863 3-953 11-269 3=-7-47 3-887 7-409 3-5-191 47-61 19-151 3=-ll-29 13=- 17 53-23 3-7-137 5-13-41 3-7-127 17-157 5-593 3-23-43 3-13-71 17-163 47-59 3-5=-37. 3^-11 5=-107 3-991 5=-7-17 13-229 3=-331 11-271 19-157 3-5-199 29-103 7=-61 3-997 41-73 5-599 3*-37 3-19-47 7-383 7-397 3' -103 lP-23 5-557 3-929 43-67 3-31= 5-577 3-5-179 3«-107 72-59 11-263 3-5-193 3=-13-23 3-7=-19 5-13-43 3-5-173 7^-53 23-113 5-7=-ll 3-29-31 3=-311 13-223 102 PRIME FACTORS Prom 3,000 to 3,100 3,100 3,200 3,200 3,300 3,300 3,400 3,400 3,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-7-11-13 5-601 31-97 3-17-59 23-131 32-5-67 7-431 3-19-53 7-443 29-107 33-5-23 13-239 3-11-97 19-179 41-83 3-5-227 32-367 5-661 5-641 3-1069 3-1103 7.11-43 7-487 32-379 3-17-61 11-283 5-7-89 3-1039 32 -'347, 5° 53-59 3-7-149 31-101 13-241 3-5-11-19 43-73 32-349 7-449 5-17-37 3-1049 47-67 23-137 3-1051 5-631 7-11-41 3^-13 29-109 132-19 33-7-17 5-643 3-5-13-17 31-107 5-683 3-17-67 13-263 11-311 3-7-163 52-137 23-149 33-127 47-73 3-29-37 ii-'293 3-52.43 7-461 3*'41 52-7-19 3-1109 52.112 3-1009 13-233 7-433 32-337 5-607 3-1613 32-359 53-61 5-647 3-13-83 41-79 7-463 3-23-47 5-11-59 17-191 32.192 3-11-101 5-23-29 47-71 32-7-53 13-257 3-5-229 7-491 19-181 3-31-37 11-313 5-13-53 32-383 17-179 3-5-7-29 11-277 3-5-223 17-197 3-1117 7-479 5-11-61 32.373 '3'- 113 43-71 5-13-47 3.1019 7-19-23 7-17-29 3-1151 5-691 3.5-7-31 3-1153 3-1087 13-251 5-653 33-112 7-467 3-1691 52-131 29-113 3-1093 17-193 72-67 32-5-73 19-173 11-13-23 3-1097 37-89 5-659 3-7-157 ' 3-1021 5-613 3.19.59 5.673 7.13.37 3.1123 3-5-211 32-5-7-11 32-11-31 37-83 7-439 3-52-41 17-181 3-7-151 19-167 52.127 32.353 11.172 3-13-89 23-151 52-139 3-19-61 72-71 592 3^-43 5-17-41 11-317 3-1163 3^.5' 11-307 31-109 3-72-23 17-199 5-677 3-1129 3-13-79 3-1061 5-72-13 5-617 32.73 3-1063 11-281 3-1031 5-619 19-163 3-1033 31-103 32-5-71 23-139 7-457 32-13-29 5-7-97 43-79 3-11-103 7-499 3-5-233 13-269 OP ODD NUMBERS. 103 Prom 3,500 to 3,600 3,600 ' 3,700 ^ 3,700 3,800 3,800 3,900 3,900 4,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 31-113 5-701 3-7-167 ll«-29 13-277 3-1201 5-7-103 7-932 3-5-13-19 11-337 3-7-181 47-83 3-1301 5-11-71 5-761 3^-47 13-293 37-103 3-31-41 5-7-109 11-347 3-19-67 3=.401 23-157 3"-5-24i 3-1303 3-1237 47-79 5-743 32-7-59 612 3.17-73 52-149 3-1171 5-19-37 7-13-43 33-5-29 3«- 17-23 7-503 13-271 3'52-47 7-11-47 3-17-71 3-1307 - 53-29 32-13-31 19-191 3-7-173 5-727 32-52-17 43-89 7-547 3-1277 52-157 3-7-11-17 3-11-113 7-13-41 . 3-11-107 5'7-'ioi 33-131 32-19-23 5-787 31-127 3-13-101 7-563 32-5-83 37-101 5-13-59 3-1279 11-349 23-167 32.7-6I 5-769 3-1213 11-331 3-29-43 19-197 5-7-107 3-1249 23-163 112-31 33-139 5-751 13-172 3-7-179 3-1181 5-709 38-5 7-521 41-89 3-1217 13-281 5-17-43 3-23-53 3-5-263 3-7-132 53-67 11-17-19 3='-5-79 3-1283 11-359 32-439 59-67 5-7-113 3-1319 37-107 17-233 3-1321 5-13-61 3-5-257 7-19-29 17-997 33-11-13 3-1187 7-509 5-23-31 3-29-41 43-83 7-523 3^-11-37 5-733 19-193 3-1223 53-71 3-5-251 5-773 3-1989 53-7-3 72-79 3-1291 53-31 34.72 11-192 99-137 3-52.53 41-97 93-173 3-1397 7-569 5-797 32-443 32-419 73-11 52-151 3-1259 3'- 397 5=^-11-13 3-1193 3-52-72 13-283 32-409 29-127 5-11-67 3-1929 7-17-31 32-431 19-199 3-13-97 5-757 7-541 32-421 17-923 11-353 3-5-7-37 . 132-23 3-5-239 17-211 37-97 3'-7-19 3-1297 17-229 5-19-41 32-433 7-557 13-307 3-113 5-17-47 7-571 3-31-43 3-1231 5-739 5-719 3-11-109 59-61 3-5-11-23 3-'-137 29- m 104 PRIME FACTORS From 4,000 to 4,100 4,100 4,200 4,200 4,300 4,300 4,400 4,400 4,500 1 3 5 7 9 11 13 15 17 19 21 23 2S 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-1367 11-373 5-821 3-37« 7-587 11-17-23 13-331 3-5-7-41 59-73 31-139 3«-479 19-227 5-863 3-1439 7-617 29-149 3-11-131 5«-173 3'- 163 7-17-37 5-881 3-13-113 3«-467 5.29« 7-601 3-23-61 3»-5-89 19-211 3-7-191 11-401 3-1471 5-883 7-631 3=-491 3»-457 5-823 23-179 3-1373 13-317 7-19-31 3-5'-ll 11-383 3-5-281 5-11-73 3-13-103 3»-7.67 41-103 5'-13« 3-1409 3'- 149 52-7-23 3-5»i59 19-233 43-103 3-7-211 11-13-31 5-887 3«- 17-29 23-193 3-17-79 29-139 37-109 3-5-269 11-367 7-577 3'- 449 13-311 5-809 3-19-71 3»- 13-37 61-71 7-619 3-5-17= 35-17 5-827 3-7-197 3-17-83 5-7-lP 19-223 3'- 157 41-101 3-1381 5-829 11-13-29 3«-461 7-593 3-1447 43-101 5-11-79- 3^-7-23 3-1481 5-7-127 3-1483 3-5-283 31-137 7-607 3-13-109 19-229 3-1451 5-13-67 3-7-193 5-811 61-73 3^-5-11 3-5-277 5-23-37 3'-ll-43 3»-ll-41 31-131 17-239 3-5-271 7^-83 13-313 3-23-59 3-1453 7^-89 73-13 3-1487 3-19-73 23-181 '5-7=-17 3=-463 11-379 43-97 3-13-107 5^-167 3-'7-i99 37-113 47-89 3'-5-31 53-79 59-71 3-11-127 7-599 5-839 3-1399 13-17-19 3-7''.29 5-853 17-251 3-1423 3=-5-97 11-397 17-257 3-31-47 5-19-47 3-1489 41-109 17-263 3^-7-71 5^-179 ll=-37 3-1493 5^-163 S'-151 7-11-53 3-1361 5-19-43 61-67 3-29-47 3'i-52-19 7-13-47 11-389 3-1427 5"- 7 3-1459 29-151 13-337 3^-487 5-877 41-107 3-7-11-19 5-857 3-1429 7-613 3<-53 5-859 3-5-13-23 7-641 3=-499 23-191 3-5-293 3^-5-7-13 17-241 5-29-31 3- 1499 11-409 3-1433 53-83 OF ODD NUMBERS. 105 Prom 4,500 to 4,600 4,600 4,700 4,700 4,800 4,800 4,900 4,900 5,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 ■ 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 7-643 3-19-79 5-17-53 43-107 3-1567 132-29 3-1601 5-312 11-19-23 3.7-229 17-283 3-5-307 17-271 11-419 3-29-53 7-659 5-13-71 3^-19 31-149 5-941 32-523 17-277 7-673 3-1571 5-23-41 . 53-89 3-112-13 32-5-109 7-701 3'- 167 13-347 3-1637 17= 5-983 3-11-149 3-5-7-43 32-5-107 61-79 3-1607 7-13-53 52-193 3-1609 11-439 3-11-137 5^-'l81 32-503 7-647 23-197 3-1511 5-907 13-349 3-17-89 19-239 7-11-59 3''-5-101 7-19-37 32-547 52-197 13-379 3-31-53 3-93-67 53-37 7-661 3-1543 11-421 41-113 3=-5-103 3' -52'- 7 99-163 3-19-83 3^-179 5-967 7-691 3-1613 47-103 29-167 3-5-17-19 37-131 13-373 32-72-11 23-211 5-971 3-1619 43-113 5-947 3-1579 7-677 11-431 32-17-31 5-13-73 47-101 3-1583 3.5.7.47 11.449 3*. 61 5". 23-43 3-17-97 72-101 3-7-13-17 5-999 3-1549 3-37-41 29-157 5-911 3-7=-31 47-97 3=-ll-47 5-72.19 72-97 3-5-317 67-71 3-13-127 5-991 3-1553 59-79 32.19-29 112-41 7-709 3-5-331 32.232 11-433 5-a53 3-7-227 19-251 13-367 3-37-43 52-191 17-281 3''-59 7-683 33 -13= 5-11-83 3-1691 5-7-139 31-157 32-541 3-5-311 13-359 7-23-29 3'- 173 3-1523 7-653 17-269 3-5=- 61 23-199 19-241 3^-509 5-'7-i3i 3-11-139 13-353 3-1657 11-443 3-5»-13 7'.17.4i 3-1697 19-957 5-977 3^-181 52-11-17 3-1559 52-199 32-7-79 13-383 17-293 3-11-151 5-997 31-151 3-7-223 5-937 43-109 3'-521 3-5-11-29 3-1663 7-93-31 ^ 3-1597 67-73 3-7-933 5-11.89 59-83 3-93-71 3-1531 5-919 13-192 3-5-313 7-11-61 37-127 5-7-137 32-13-41 3'-5-37 19-963 3?- 7- 73 — 1 14 106 PRIME FACTORS Prom 5,000 to 5,100 5,100 5,200 5,200 5,300 5,300 5,400 5^00 5,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-1667 5- 7-' ii- 13*' 3-1669 7-743 ll''-43 3-5-347 41-127 3'- 19-31 11-491 3-1801 5-23-47 3«-7 5-1021 5-1061 3-29-61 3-13-131 19-269 s-'s-ii-'si" 7-17-43 32- 601 7-773 33-193 13-401 5-7-149 3-37-47 17-307 23-227 3-1741 5=-ll-19 47-113 3-7-11-23 5-1063 13-409 33-197 17-313 3= -557 5-17-59 29-173 3-7-239 3-5-192 3^-569 47-109 53-41 3-1709 23-223 7-733 3-29-59 5-13-79 11-467 3^-571 53-97 37-139 3-5-73 3-13-139 11-17-29 52-7-31 3"- 67 61-89 3-5''-67 11-457 47-107 3^-13-43 7-719 5-19-53 3-23-73 3-5=.71 7-761 73= 3-1777 3=-7-83 3-1811 5-1087 3-5-349 5-11-97 3=-593 19-281 7= -109 3-13-137 5-1069 132-31 3-1747 7^-107 5-1049 32-11-53 29-181 59-89 3-17-103 5-1051 7-751 3-1753 3-72.37 7P 3-41= 5-1009 7^-103 33-11-17 32-5-112 13-419 19-271 3-17-101 5-1631 33-191 7-11-67 13-397 3-1721 5-1033 3-1783 53-i6i 32-5-7-17 11-487 23-233 3-1787 31-173 5-29-37 3-1789 7-13-59 41-131 33-199 53-43 19-283 3-11-163 3-23-79 7-19-41 5-1091 3-17-107 53-103 43-127 32- 607 5-1093 7-11-71 3-1823 31-163 3-5-337 13-389 3-7-241 61-83 5-1013 3^-563 37-137 11-461 3-19-89 19-277 3^-5-13 23-229 11-479 3-7-251 3-1723 ' 7-739 3'i-5='-23 31-167 13-421 3-52.73 5'i-211 3-1759 3-1693 i3-i7>23' ' '.' 3^-5-113 7-727 3-1697 11-463 5-1019 3-1699 3-11-157 71-73 5-17-61 3-7-13-19 29-179 3= -577 5-1039 33-7-29 3' -587 5-7-151 17-311 3-41-43 11-13-37 67-79 3-5-353 7-769 3-5-359 17-317 32-599 5-1097 3-31-59 11-499 172.19 3-1831 5-7-157 23-239 32-13-47 5-13-83 3-7-257 3-1733 7-757 OF ODD NUMBERS. 107 From 5,500 to 6,600 5,600 5,700 5,700 5,800 5,800 5,900 5,900 6,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37. 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-1867 13 431 5-19-59 3'-7-89 71-79 31-181 3-1871 5-1123 41-137 3-1873 7-11-73 3-1901 5-7-163 13-439 3-11-173 29-197 32-5-127 7-'i9-43 3-1907 59-97 52-229 3-23-83 17-337 11-521 32-72-13 5-31-37 7-899 32-5-43 37-i57 3-13-149 3-7-281 3-5-367 5-1181 3-11-179 19-311 23-257 3*-73 5-7-132 61-97 3-1973 31-191 7-787 3-11-167 37-149 5-1103 3'i-613 5-1163 3-7-277 11-232 3-7-263 52-13-17 32-647 52-233 3'-5* 17-331 13-433 3-1877 43-131 5-7'-23 3-1879 3-52-79 3-19-97 3-29-67 v-n 19-307 3-5-389 13-449 72-112 32-659 17-349 5-1187 3-1979 11-503 3^-5-41 7- -113 29-191 3-1847 23-241 5-1109 3-43' 31-179 7-13-61 3'-617 5-11-101 3-i7-'lb9""' 67-83 3-1913 32-11-59 13-457 3-7-283 5-29-41 19-313 32- 661 11-541 3=-ll-19 5-1129 3-5-383 7-821 5-7-167 3-1949 3-7-269 3*-71 11-523 5-1151 3-19-101 13-443 7-823 3-17-113 5-1153 73-79 32-641 29-199 23-251 3-52-7-11 53-109 3-195* 5-1171 3^ -7- 31 3-5-13-29 3-5-397 7-23-37 59-101 3-1987 67-89 5-1193 3^-13-17 47-127 7-853 3-11-181 52-239 43-139 3-1993 3'-17-37 7-809 5-11-103 3-1889 11-13-41 3-5-17-23 3-5-7-53 19-293 3' -619 5'-223 3-11-13^ 7-797 53-107 3-31-61 5' -227 7-811 32-631 13-19-23 3-19-103 7-839 53-47 32-653 3-41-47 3-1861 5-1117 37-151 3^-23 7-i7-47 3-5-373 29-193 11-509 3-37-53 5-11-107 7-292 3-13-151 43-137 71-83 32-5-131 31-193 32-5-7-19 3-5-379 112-47 5-13-89 32-643 7-827 53-113 3-1997 13-461 5-11-109 3-1999 7-857 r' 3-7-271 3-1931 5-19-61 11-17-31 3-1933 5-17-67 33-211 41-139 17-347 108 PRIME FACTORS From 6,000 to 6,100 6,100 6,200 6,200 6,300 6,300 6,400 6,400 6,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 17-353 3='23-29 5-1201 3-2003 7-859 3-5-401 11-547 13-463 3^-223 19-317 52-241 3-7''-41 37-163 3-2011 5-17-71 3'-ll-61 7-863 3-5-i3"-3i' 23-263 "" 3-2017 s-i-'m'" 3' -673 73-83 11-19-29 3-43-47 5-1213 3-7-17* 13-467 36.52 59-103 3-2027 7-11-79 5-1217 3-2029 3^-677 5-23-53 7-13.67 3-19-107 17-359 3-5-11-37 31-197 41-149 3'-7-97 5-1223 *"" 3-2039 29-211 3.13-157 53.72 11-557 3^-227 3-5-409 17-19^ 7-877 3-23-89 5-1229 32-683 11-13-43 3-7-293 5-1231 47-131 3-2053 61-101 3=-5-137 7-881 31-199 3-1P-17 5^-13-19 3-29-71 37-167 7-883 3'-229 5-1237 23-269 3-2063 41-151 11-563 3-5-7-59 32-13-53 5-17-73"" 3-2069 7-887 3-19-109 5-11-113 32-691 72-127 3-52-83 13-479 3-31-67 23-271 5-29-43 3*-7-ll 17-367 792 3-2081 5-1249 3-2083 7-19-47 132 -.37 32-5-139 'li-569' 3-2087 5-7-179 3-2089 32-17-41 52-251 3-7-13-23 11-571 61-103 3-5-419 19-331 3'-233 7-29-31 5-1259 3-2099 3-11-191 5-13-97 7-17-53 32-701 59-107 3-5-421 71-89 3-72-43 52-11-23 32-19-37 13-487 3-2111 5-7-181 3-2113 17-373 3'-5-47 11-577 7-907 3-29-73 5-31-41 3-13-163 32-7-101 5-19-67 3-11-193 23-277 37-173 19-337 3-5-7-61 43-149 13-17-29 3-2137 112-53 5-1283 32-23-31 72-131 3-2141 52-257 3-2143 59-109 7-919 32-5-11-13 41-157 47-137 3-19-113 17-379 5-1289 3-7-307 3'-239 5-1291 11-587 3-2153 7-13-71 23-281 3-5-431 29-223 32-719 3-5'-17 7-911 32-709 13-491 5-1277 3-2129 7-11-83 3-2131 5-1279 3^-79 52-7-37 3-17-127 11-19-31 3-2161 5-1297 13-499 32-7-103 43-151 3-5-433 73-89 67-97 OP ODD NUMBERS. 109 From 6,500 to 6,600 6,600 6,700 6,700 6,800 6,800 6,900 6,900 7,000 1 3 5 7 9 XI 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-11-197 7-929 5-1301 33 -941 23-283 17-383 3-13-167 5-1303 73-19 3-41-53 11-593 32-5''-29 61-107 3-7-311 47-139 5-1307 3-2179' 13-503 31-211 3^-727 5-7-11-17 3-37-59 3-5-19-23 79-83 7-937 38 5-13-101 3-11-199 3-7-313 5^-263 3^-17-43 29-227 3-5-439 7-941 11-599 3-133 19-347 5-1319 3^^-733 7-23-41 3-31-71 5-1321 3-2203 11-601 17-389 33-5-72 13-509 3-2207 37-179 53-53 3-472 7-947 19-349 32-11-67 5-1327 3-2213 29-229 7-13-73 3-5-443 172-23 61-109 32-739 5-113 3-7-317 3-2221 5-31-43 59-113 33-13-19 7-953 3-52-89 11-607 3-17-131 41-163 5-7-191 32-743 3-23-97 5-13-103 37-181 3-7-11-29 32-5-149 19-353 3-2237 72.137 5-17-79 3-2239 11-13-47 3" -83 52-269 7-312 3-2243 53-127 3-5-449 23-293 32-7-107 11-613 5-19-71 3-13-173 17-397 43-157 3-2251 5-7-193 29-233 32-751 3-5-11-41 67-101 7-967 3-37-61 13-521 52-271 33-251 3-7-17-19 5-23-59 11-617 3-31-73 32-5-151 7-971 13-523 3-2267 5-1361 3-2269 11-619 72-139 32-757 5-29-47 17-401 3-2273 19-359 3-52-7-13 33-11-23 5-1367 3-43-53 7-977 3-2281 5-372 41-167 32-761 13-17-31 7-11-89 3-5-457 193 3-2287 5- 1373' 32-7-109 3-29-79 5''-ll 13-232 3-2293 7-983 3^-5-17 71-97 832 3-2297 61-113 5-7-197 3-112-19 67-103 32-13-59 5-1381 3-72-47*" 31-^23 3-5-461 '11-17-37' 32-769 7-23-43 52-277 3-2309 132.41 29-239 3-23U 5-19-73 7-991 33-257 11-631 53-131 3-5-463 3-7-331 17-409 5-13-107 32-773 3-11-2H 5-7-199 3'-23-'ib"l' 19-367 32.52-31 7-997 "" 3-13-179 5-11-127 3-17-137 29-241 33-7-37 5-1399 3-2333' 110 PRIME FACTORS From 7,000 to 7,100 7,100 7,200 7,200 7,300 7,300 7,400 7,400 7,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 47-149 3-5-467 7^-11-13 43-163 3-19-41 5-23-61 3-2339 7-17-59 3-2341 5''-281 S'-ll-?! 79-89 13-541 3-5-7-67 31-227 3-2347 5-1409 3^-29 7-19-53 11-641 3-2351 5-17-83 3-13-181 23-307 7-1009 32-5-157 37-191 3-2357 11-643 5^-283 3-7-337 3^-263 5-7^-29 3-23-103 73-97 3=-787 5-13-109 19-373 3-17-139 7-1013 41-173 3-5-11-43 47-151 31-229 13-547 3-2371 5-1423 11-647 32-7-113 17-419 3-53-19 3-2377 7-1019 5-1427 32-13-61 112-59 37-193 3-2381 5-1429 7-1021 3-2383 23-311 33-5-53 17-421 3-7-11-31 13-19-29 5-1433 3-2389 67-107 71-101 32-797 52-7-41 3-2393 43-167 11-653 3-5-479 7-13-79 32-17-47 5-1439 " 3-2399 23-313 19-379 3-7^ 5-11-131 3''-89 3-5-13-37 7-1031 3-29-83 31-233 52-172 32-11-73 7-1033 3-2411 5-1447 3-19-127 13-557 32-5-7-23 ii-659 3-2417 5-1451 3-41-59 7-17-61 53-137 3'-269 5-1453 132-43 3-2423 11-661 7-1039 3-52-97 19-383 29-251 32-809 s'.si'.'ii"" 3-7-347 37-197 23-317 3-11-13-17 5-1459 72-149 67-109 3-5-487 3-2437 71-103 5-7-11-19 33-271 13-563 3-2441 52-293 17-431 3-7-349 32-5-I63 11-23-29 41-179 3-2447 7-1049 5-13-113 3-31-79 32-19-43 5-1471 7-1051 3-11-223 17-433 37-199 3-5-491 53-139 3*-7-13 73-101 53-59 3-2459 47-157 112-61 3-23-107 5-7-211 83-89 32-821 19-389 32-811 3-5-17-29 13-569 72-151 3-2467 11-673 5-1481 32-823 31-239 3-7.353 5-1483 3-2473 41-181 13-571 33-52-11 7-1061 17-19-23 3-2477 5-1487 3-37-67 43-173 7-1063 32-827 5-1489 11-677 3-13-191 29-257 ■ 3-5-7-71 32-829 17-439 5-1493 3-19-131 7-11-97 31-241 3-47-53 52-13-23 33-277" "" 7-1069 3-5-499 3-11-227 59-127 5-1499 32-72-17 OP ODD NUMBERS. Ill From 7,500 to 7,600 7,600 7,700 7,700 7,800 7,800 7,900 7,900 8,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 13-577 3-41-61 5-19-79 11-691 3-17-151 29-269 5-7-223 37-211 3-19-137 73-107 13-601 3-5-521 ■T-iiif*"""" 3''-ll-79 5^ -"313" 3-2609 7-1129 3-5-17-31 3^-5-13= 5-23-67 3-7-367 13-593 11-701 3''-857 5-1543 3- si- 83""'" 7-1103 s-h'-'m"" 3-2503 7-29-37 11-683 3''-5-167 7-1087 3-43-59 23-331 5-1523 3-2539 19-401 11-719 33-293 41-193 5-1583 3-7-13-29 73-103 3-23-109 89= 3-19-139 5'- 317 3^-7-112 53-61 "29-263 3-2543 13-587 17-449 3-5-509 7-1091 5'- 7- 43 '3-13-193 59-131 3«-859 11-19-37 5-7-13-17 3-2579 71-109 3^-881 7-11-103 17-443 3=' -31 5-11-137 41-191 3-7-373 5-1567 17-461 3'- 13- 67 3-5-23= 3-7-359 17-,467 3-2647 13= -47 5-7-227 3=-883 3^-283 s-ii-'isg'"" 3-2549 19-397 3-5-503 3-29-89 5-1549 61-127 3^-7-41 23-337 3"-5-"ii-47"" 11-23-31 3-5-523 7-19-59 47-167 3-2617 5-1571 3" -97 29-271 7-1123 3-2621 5- IF- 13 3= -839 7-13-83 5-1511 3-11-229 7-1093 3-2551 ' 5-1531 13-19-31 3^-23-37 47-163 79-97 3-5-7-73 11-17-41 3-11-241 5-37-43 73-109 3-7-379 19-419 3-13-199 7-1109 5-1553 3^-863 17-457 19-409 3-2591 5=-311 7-11-101 3-2593 31-251 43-181 3=-5-173 13-599 3-2521 5-17-89 7-23-47 67-113 3'-5-59 31-257 13-613 3-2657 7-17-67 5=- 11-29 3-2659 79-101 23-347 3^-887 5-1597 7=-163 3-2663 61-131 3-43-61 17-463 3-2557 3-5='-101 5''-307 3^-853 7-1097 3'-53-7 11-13-53 3-7-]9« 5-'-37-4i 3^-281 3-37-71 3-13-197 5-29-53 5-19-83 3-11-239 7^-23 13-607 3'- 877 5-1579 53-149 3-2633 3-11-233 3-72-53 3-2531 5-7«-31 71-107 3-17-149 7" -157 3*-5-19 43-179 5-1559 3-23-113 11-709 3-5-13-41 11-727 19-421 112 PRIME FACTORS From 8,000 to 8,100 8,100 8,200 8,200 8,300 8,300 8,400 8,400 8,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 53-151 5-1601 3-17-157 3-2671 • 5-7-229 3«-ll 13-617 71-113 3-5«-107 23-349 7-31-37 3-2677 29-277 5-1607 3=^-19-47 11-17-43 3-7-383 5-1609 13-619 3-2683 83-97 3^^-5-179 7-1151 3-2687 11-733 5-1613 3-2689 7-1153 3'- 13-23 5=-17-19 41-197 3-2693 59-137 3-5-7^-11 3^-29-31 5-1619 3-2699 7-13-89 3-37-73 5-1621 ll«-67 3«-17-53 7-19-61 3-5-541 23-353 3-2707 5"- 13 33-7-43 11-739 47-173 3-2711 5-1627 79-103 3-2713 7-1163 17-479 3^-5-181 29-281 3-11-13- 31-263 5-7.233 3-2719 41-199 19 3^-907 5-23-71 3-7-389 11-743 3-5*^-109 13-17-37 3^-101 72-167 5-1637 3-2729 19-431 3-2731 5-11-149 7-1171 3^-911 59-139 13-631 3-5-547 29-283 3-7-17-23 43-191 5-31-53 32-11-83 3-2741 5'-7-47 19-433 3-13-211 33-5-61 7-11-107" 3-41-67 5-17-97 3-2749 73-113 37-223 32-7-131 5-13-127 23-359 3-2753 11-751 3-5-19-29 7-1181 32-919 52-331 3-31-89 17-487 72-132 3-11-251 52-331 33-307 3-5-7-79 43-193 ' 3-2767 192-23 5-11-151 32-13-71 7-1187 3-17-163 5-1663 3-47-59 53-157 7-29-41 32-52-37 11-757 3-2777 13-641 5-1667 3-7-397 31-269 19-439 3^-103 5-1669 17-491 3-112-23 7-1193 3-5-557 61-137 13-643 32-929 5-7-239 3-2789 11-761 3-2791 53-67 32-72-19 172-29 83-101 3-5-13-43 3-2797 7-11-109 5-23-73 33-311 37-227. 31-271 3-2801 5-41» 7-1201 3-2803 13-647 47-179 32-5-11-17 19-443 3-7-401 52-337 3-532 32-937 5-7-241 11-13-59 3-29-97 23-367 ■3-'5-563" 7-17-71 33-313 79-107 5- 19-89 3-2819 11-769 3-7-13-31 5-1693 3v"94l" 43-197 37-229 3-52-113 72-173 61-139 3-11-257 17-499 5-1697 32-23-41 13-653 7-1213 3-19-149 5-1699 29-293 3-2833 OP ODD NUMBERS. 113 From 8,500 to 8,600 8,600 8,700 8,700 8,800 8,800 8,900 8,900 9,000 i 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 11-773 3»-5>7 47-181 67-127 3-9837 3-47-61 7-1229 5-1721 3-19-151 7-11-113 32-967 5-1741 13-677 33-23-43 29-307 5-13-137 ' 3-2969 59-151 7-19-67 3-2971 5-1783 37-241 32-991 11-811 3-5-587 3-2903 31-281 23-383 32-11-89 7-1259 5-43-41 3-2939 79-109 33- 11-29 5-1723 7-1231 3-132-17 37-233 5-13-131 3-17-167 7-1217 3-5-7-83 23-379 3'-17-19 11-13-61 52-349 3-2909 7-29-43 3= -947 52-11-31 3-17-173 52-353 7-13-97 3^-109 3-5'-23 3-52-7-17 79-113 3-2843 19-449 7-23-53 3-5-569 32-7-137 89-97 5-11-157 3-2879 53-163 3-'43-67 5-7-13-19 3-13-229 3-41-71 5-1747 112-73 3-5-19-31 5-1787 33-331 7-1277 32-971 7-1249 3-5-11-53 3=-13-73 3-7-421 37-239 5-29-61 32-983 53-167 3-13-227 5-7-11-23 17-5?1 3-2953 3-11-271 5-1789 23-389 3-19-157 5-1709 3-7-11-37 83-103 17-503 3-2851 5-29-59 43-199 3'-317 7-1223 32-312 41-211 17-509 3-5-577 11-787 7-1237 , 3-2887 13-673 3-2917 h'-ii'-wi"" 32-7-139 19-461 7-1279 32-5-199 132-53 172-31 3-29-103 3-23-127 5-1753 11-797 3-37-79 72.179 31-283 33-52-13 67-131 3-5-571 13-659 11-19-41 3-2857 5-1733 30-107 32-5-197 72-'l81 3-2957 19-467 53-71 3-11-269 13-683 83-107 33-7-47- 5-1777 s-ii-ies ' 3- 72- 61 13-23-29 3-72-59 52-347 32-997 52-359 47-191 3-41-73 7-1283 13-691 3-5-599 11-19-43 89-101 3^-37 17-232 5-7-257 3-2999 52.73 32-953 23-373 3-11-263 3-2927 3-2861 5-17-101 31-277 3-7-409 11=- 71 13-661 32-5-191 19-457 - 32-5-193 7-17-73 5-7-251 3-29-101 11-17-47 59-149 3^-977 5-1759 , 19-463 3-7-419 3-2963 17-523 3-2897 5-37-47 3-13-223 3-5-593 7-31-41 11-809 15 114 PRIME FACTORS From 9,000 to 9,100 9,100 9,200 .9,200 9,300 9,300 9,400 9,400 9,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-3661 5-1801 19-479 3-3067 71-131 3-7-443 5-1861 41-227 3-29-107 7-17-79 s'-'s-'i'i'.'ig" 23-409 972 3-3137 3-5-607 7-1301 3-3037 13-701 5-1823 32-1013 11-829 7-1303 3-3041 53-73 5-7-263 33-U-3I 61-151 3-37-83 5-19-97 13-709 3-7-439 32-7-11-13 67-139 ,3<-5-23 7-113 3-5-601 71-197 29-311 3-31-97 7-1289 52-192 32-17-59 11-821 3-3011 5-13-139 7-1291 3-23-131 5-7-269 3-43-73 ■ 3-13-239 33'-'3'4'9 52-13-29 11-857 3-7-449 23-401 32-52-41 52-373 3-3109 19-491 7-31-43 32-17-61 5-1867 3-17-179 23-397 11-839 3-17-181 7-1319 5-1847 3-3079 32-5-7-29 3-5-17-37 13-19-37 3-11-277 41-223 5-31-59 3-3049 ' 7-1307 3-11-283 ■ 32-1049 7-19-71 5-1889 3-47-67 11-859 13-727 3-23-137 5-31-61 72-193 32-1051 32-13-79 5-432 7-1321 3-3083 11-292 19-487 3-5-617 3-5-7-89" 13-719 33-5-67 83-109 3-7-431 11-823 5-1811 3-3019 13-17-41 32-19-53 5-72-37 32-1039 47-199 5-1871 3-3119 72-191 11-23-37 3-3121 5-1873 17-19-99 33-347 7- 13-163"'"' 3-5' 3^-113 5-1831 3-43-71 47-197 33-73 59-157 5-17-109 3-3089 13-23-31 73-127 3-11-281 52-7-53 72-11-17 3-5-13-47 89-103 53-173 32-1019 3-5-631 3-3023 47-193 43-211 3-52-112 29-313 7-1297 32-1009 31-293 5-23-79 3- 13-933 61-149 17-557. 3-7-11-41 52-367 3-7-19-93 67-137 52-379 3»-13 32-1031 83-113 3-53-59 11-853 5-1877 32-7-149 41-929 19-499 3-29-109 5-7-271 53-179 3-3163 3-3061 5-11-167 3-5-619 37-251 7-1397 3-;9-163 5-]i-i3^""' 32-1033 17-547 32-1021 7-13-101 99-317 3-5-613 17-541 3-7-433 5-17-107 11-827 33-337 3-31-101 5-1879 11-863 32-5-211 3-13-241 7-23-59 OP ODD NUMBERS. 115 From 9,500 to 9,600 9,600 9,700 9,700 9,800 9,800 9,900 9,900 10,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-3167 13-17-43 5-1901 3-3169 37-257 3=-7-151 5-11-173 31-307 3-19-167 89-107 3-5^-127 7-1361 13-733 3' -353 5-1907 3-11-172 7-29-47 3-31-81 5-23-83 3^-1061 41-233 3-5-72-13 19-503 11^-79 3-3187 73-131 5-1913 32-1063 7-1367 17-563 3-3191 52-383 61-157 3-31-103 11-13-67 7-372 33-5-71 43-223 3-23-139 53-181 5-19-101 3-7-457 29-331 32-11-97 5-17-113 13-739 3-3203 7-1373 3-5-641" 59-163 32-1069 5'-7-ll 3-3209 3-132-19 5-41-47 23-419 3''-7-17 31-311 3-5-643" 11-877 3-3217 72-197 5-1931 32-29-37 13-743 3221 1933 1381 11-293 19-509 17-569 32 ••52 -43 3-7-461 23-421 5-13-149 3-3229 11-881 33-359 5-7-277 89-109 31-313 3-5-647 17-571 7-19-73 32-13-83 11-883 5-29-67 3-41-79 3-7-463 52-389 71-137 32-23-47 37-263 3'-5-ii'-59" 7-13-107 3-17-191 5-1949"' 33-192 72-199 3-3251 5-1951 11-887 3-3253 43-227 13-751 32-5-7 31 3-3257 29-337 52-17-23 3-3259 7-11-127 32-1087 5-19-103 3-13-251 3-53-61 7-1399 3-5-653 97-101 41-239 3^-112 5-37-53 3-7-467 17-577 3-3271 5-13-151 32-1091 7-23-61 11-19-47 3-52-131 31-317 3-29-113 5- 7-281" 32-1093 13-757 3-17-193 5-11-179 43-229 32-72-67 59-167 33-5-73 3-19-173 7-1409 5-1973 3-11-13-23 71-139 32-1097 53-79 7-17-83 3-37-89 41-241 3-5-659 n-29-3i' 32-7-157 13-761 5-1979 3-3299 19-521 3-3301 5-7-283 33-367 11-17-53 23-431 3-5-661 47-211 7-13-109 3-3307 52-'397" 32-1103 3-7-11-43 5-1987 19-523 3-3313 61-163 32-5-13-17 73-29 3-31-107 37-269 5-11-181 3-3319 23-433 7-1423 3= -41 5-1993 3-3323 " 132-59 3-52-7-19 11-907 17-587 3' -1109 67-149 5-1997 3-3329 7-1427 97-103 3-3331 5-1999 13-769 32-11-101 116 PRIME FACTORS From 10,000 to 10,100 10,100 10,200 10,200 10,300 10,300 10,400 10,400 10,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 73-137 7-1429 3-5-23-29 3-7-13-37 5-'43-47 3^-1123 11-919 3-3371 5-7-17= 67-151 3-3373 29-349 53-191 3' -5' 13-19-41 7-1447 3-11-307 1012 3-19-179 5-13-157 59-173 3-41-83 7-1459 32-5-227 17-601 11-929 3-3407 52-409 3-7-487 53-193 13-787 3''-379 5-23-89 29-353 3-3413 72.11-19 3-3467 101-103 5-2081 3-3469 7-1487 29-359 S'- 13-89 5-2083 11-947 3-23-151 17-613 7-1489 3-5«.139 32-5-229 11-937 132. 61 3-7-491 3-47'71 17-19-31 5-2003 3^-7-53 43-233 11-911 3-13-257 5= -401 37-271 3-3343 7-1433 79-127 3=-5-223 5-2063 3-19-181 17-607 32.31-37 52-7-59 23-449 3-11.313 32.I9.6I 5-2027 3-31-109 3.5-13-53 72-211 33-383 5-2087 3.72.71 11-13.73 53-197 3-592 5-2089 31-337 3».43 7.1493 3-3347 lP-83 S-7»-41 3-17-197 13-773 19 -23^ 3"- 1117 5-2011 89-113 3-7-479 32-72-23 5-2029 73-139 3-17-199 3-5-683 5-2069 3-3449 79-131 11-941 3-7-17-29 5-19-109 37-277 32-17-67 ii-i3-7i 3-5-677 7-1451 5-7-293 3-13-263 3.5-17-41 32-1151 13-797 43-241 3-5-691 7-1481 3«-1129 31-331 3-11-311 5-2053 3-11-317 29-347 3-5-11-61 5-19-107 3-3389 5-7-13-23 32-1163 192-29 37-283 3-3491 52-419 32-7-163 3' -373 7-1439 5^-13-31 3-3359 7-1453 3-3391 52-11-37 3-3457 11-23-41 5^-83 3»-1153 97-107 7-1483 3-3461 5-31-67 13-17-47 3-3461 19-547 3'-5.7.11 37-281 3-52-137 43-239 19-541 3-23-149 7-13-113 5-112-17 3*- 127 3^-13-29 17-599 3-5-7-97 61-167 23-443 3-43-79 3-7-499 47-223 11-953 32.5.233 17-593 3-3361 5-2017 7-11-131 3»- 19-59 17.617 3.13.269 7-1499 5-2099 3-3499 41-251 3-47-73 5-29-71 7-1471 3-3433 3-5-673 23-439 5-2039 32-11-103 7-31-47 OF ODD NUMBERS. 117 Prom 10,500 to 10,600 10,600 10,700 10,700 10,800 10,800 10,900 10,900 11,000 X 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 "37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3'-389 5-11-191 7-19.79 3-31-113 23-457 3-5-701 13-809 67-157 3=-7-167 17-619 52.421. 3-1P-29 3-3511 5-7=-43 41-257 3=-1171 83-127 13-811 3-5-19-37 53-199 7-11-137 3-3517 61-173 5-2111 33-17-23 59-179 3-7-503 5-2113 3- 13-271* 11-31= 97-109 3«-5''-47 7-1511 71-149 3-3527 19-557 5-29-73 3-3529 7-17-89 5-13-163 23-461 3-5-7-101 io^' 3<-131 5-11-193 3-3539 7-37-41 13-19-43 3-3541 5<-17 3= -1181 73-31 3-5-709 11-967 3-3547 29-367 5-2129 32-7-13= 23-463 3-5SI-67 5-2131 3-11-17-19 7-1523 33-5-79 47-227* * 3-3557 13-821 5=-7-61 3 -3359 59-181 11-971 3=-1187 5-2137 3-7-509 3-3533 172-37 3-5-23-31 19-563 13-823 32-29-41 7-11-139 5-2141 3-43-83 3-3571 5-2143 7-1531 33-397 71-151 3-'52-'ll. 17-631 13 3-72-73 s-'ig-iis' 32-1193 23-467 3-3581 5-7-307 11-977 3-3583 13-827 32-5-239 31-347 7-29-53 3-17-211 47-229 5-3153 3-37-97 112-89 3^-7-19 52-431 13-829 3-3593 41-263 3-5-719 7-23-67 32-11-109 43-251 5-17-127 3-59-61 7-1543 3-13-277 5-2161 101-107 32-1201 19-569 11-983 3-5-7-103 29-373 31-349 3-3607 79-137 52-433 33-401 72-13-17 3-23-157 5-11-197 3-3613 37-293 7-1549 33.5-241 i9-57i'*' 3-3617 5-13-167 3-7-11-47 32-17-71 5-41-53 3-3623 7-1553 83-131 3-53-29 73-149 11-23-43 33-13-31 5-7-311 3-19-191 3-3631 5-2179 17-641 32-7-173 11-991 3-5-727 13-839 3-3637 7-1559 5-37-59 32-1213 61-179 67-163 3-11-331 52-19-23 72-223 3-3643 17-643 13-292 3'-5 3-7-521 31-353 5-11-199 3-41-89 47-233 32-1217 5-7-313 3-i3-28i * 97-113 19-577 3-5-17-43 11-997 7-1567 32-23-53 52-'439"** 3-3659 79-139 3'7-523 5-133 33-*n-'37 * 29-379 3-5-733*' 7-1571 17-647 118 PRIME FACTORS From 11,000 to 11,100 11,100 11,200 11,200 11,300 11,300 11,400 11,400 11,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-19-193 5-3i-7i 3«-1223 101-109 7-ll=-13 3-3671 5-2203 23-479 3-3673 103-107 73-151 32.52.7!! 17-653 3-3701 5-2221 29-383 3-7-23« 41-271 23-487 17-659 3»-5-83 7-1601 11-1019 3-37-101 3-3767 89-127 5-7-17-19 3-3769 43-263 13-877 3''-7-181 5-2281 11-17-61 3-3803 33-419 5-31-73 101-113 3-5-761 7»-233 19-601 3' -47 3'-5-13-19 5-2243 3-3739 13-863 7= -229 3'-29-43 5= -449 103-109 3-19-197 11-1021 47-239 3-5-7-107 17-661 3-7=- 11 3-11-337 7»-227 5^-89 3-3709 31-359 13=- 67 3-5=-151 47-241 5=-457 3-13-293 11-1039 7-23-71 3-37-103 5-2287 41-269 3-3677 11-17-59 5-2207 3-13-283 7-19-83 61-181 3^-409 5-47^ 3'i-1259 7-1619 5-2267 3-3779 17-23-29 11-1031 3-19-199 5-2269 7-1621 3'-13-97 3^-1237 5-17-131 7-37-43 3-47-79 13-857 11-1013 3-5-743 71-157 3'-31-41 17-673 3= -1249 5-13-173 3-23-163 7-1607 h'-iv-ii"" 5-2251 3-5-7-109 , ior 3-11-347 13-881 5-29-79 3»-19-67 7-1637 73.157 3-3821 5-2293 3-29-127 43-257 7-1579 3-5-11-67 3'-7-59 19-587 5-23-97 3-3719 3-5-757 41-277 37-307 3-7-541 11-1033 5-2273 3' -421 3''-139 3=-1229 13-23-37 5-2213 3-7-17-31 3-6P 5-7-11-29 13-859 3'-17-73 7-1609 3-5-751 19-593 59-191 3-13-17= 3-3823 ' 83-137 3-17-223 5»-7-13 31-367 3-3793 19-599 3-3691 52-443 11-19-53 3^-1231 7-1583 7-11-149 3»-5«-17 23-499 13-883 3-43-89 3-5^-149 5«-ll-41 3«-7-179 7-1597 3-3727 53-211 5-2237 3''-ll-113 67-167 19'-31 3-7-13-41 5-2239 29-389 3-3761 5-37-61 3-5-739 3^-5-11-23 59-193 7-1627 3-3797 5-2297 3-7-547 13-853 3-3697 5-7-317 3'- 137 11-1009 3-53-71 7-1613 23-491 3«-5-251 11-13-79 3«-1277 5-ll«-19 5-43-53 3-29-131 3-3733 3-3833 OP ODD NUMBERS. 119 From 11,500 to 11,600 11,600 11,700 11,700 11,800 11,800 11,900 11,900 12,000 1 3 5 7 9 11 13 IS 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 7-31-53 3-5-13'59 37-311 17-677 3= • 1279 29-397 5-7''-47 3-11-349 41-281 3-23-167 5^-461 3'-7-61 13-887 19-607 3-5-769 83-139 11-1049 3-3847 7-17-97 5-2309 3' -1283 3-3851 5-2311 7-13-127 3-3853 11-1051 31-373 3'-5-257 43-269 23-503 3-7-19-29 71-163 5=-463 3-17-227 37-313 3''-ll-13 5-7-331 3-3863 67-173 3-5-773' 7-1657"' 3^-1289 41-263 5-11-211 3-53-73, 13-19-47 17-683 3- 7'- 79 5-23-101 3'i-1291 59-197 3-53-31 7-11-151 29-401 3-3877 5-13-179 3^-431 103-113 7-1663 3-3881 5-17-137 19-613 3-11-353 61-191 43-271 3''-5-7-37 89-131 3-13^-23 107-109 5-2333 3-3889 7-1667 11-1061 3^-1297 5=-467 3-i7-'229 7-1669 3-5-19-41 13-29.31 3= -433 ■ 11-1063 5-2339 3-7-557 3-47-83 5-2341 23-509 3^-1301 7'-239 13-17-53 3-5-11-71 3-3907 19-617 5'- 7- 67 3^-1303 37-317 3-3911 5-2347 11=- 97 3-7-13-43 59-199 3*-5-29 11-691 31-379 3-3917 7-23-73 5-2351 3-3919 11-1069 19-619 3''- 1307 5-13-181 7-4P 3-3923 79-149 61-193 3-5«-157 3''-7-ll-17 5-2357 3-3929 13-907 3-3931 5-7-337 47-251 3^-19-23 11-29-37 3-5-787 7^-241 3-31-127 5- 17-139" 3^-13-101 53-223 3-7-563 5=^-11-43 3-3943 3^-5-263 7-19-89 3-3947 13-911 5-23-103 3-11-359 17==- 41 7-1693 3' -439 5-2371 71-167 3-50-67 29-409 3-5-7-113 11-13-83" 32-1319 31-383 5'' -19 3-37-107 7-1697 109== 3-17-233 5-2377 3^-1321 11-23-47 7-1699 3-5-13-61 2-3967 5-2381 36.72 43-277 3-11-19'' 5-2383 17-701 3-29-137 7-13-131 s^-'s'^i'-ss " 3-41-97 5-7-11-31 3-23-173 3=-1327 5-2389 13-919 3-7-569 17-19-37 3-5-797" 11-1087 3'- 443 7-1709 5-2393 3-3989 3-13-307 5= -479 7-29-59 3^- IP 23-521 3-5-17-47 73-163 19-631 3-7-571 67-179 5-2399 3^-31-43 120 PRIME FACTORS Prom 12,000 to 12,100 12,100 12,200 . 12,200 12,300 12,300 12,400 12,400 12,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 11-1091 3-4001 5-7* 3-4663 41-293 3»-5-89 61-197 7-17-101 3-4007 11-1093 52-13-37 3-19-211 23-523 53-227 3^-7-191 5-29-83 3-7=-83 72-13-19 3«-5-269 32-1367 5-23-107 31-397 3-11-373 13-947 7-1759 3-5-821 109-113 97-127 32-372 52-'lV-29 " ' " ' 3-7-587 79-157 3-5-827 19-653 5-2441 3-13-313 29-421 3-11-367 5-2423 3-7-577 32-7-197 32-23-59 5-7-349 19-643 3-4073 IP- 101 17-719 3-5=- 163 7-1747 3*- 151 13-941 5-2447 3-4079 5-13-191 3-4139 11-1129 17-23-31 3' -449 5'- 97 67-181 3-13-311 7-1733 11-1103 3-5-809 53-229 61-199 3'!-19-71 3-41-101 52-7-71 172-43 32-1381 31-401 11-19-59 3-4111 5-2467 132-73 3^-457 7-41-43 3-5-839 3-4013 7-1777 3-11-13-29 23-541 5-19-131 3»-461 59-211 • 3--7-'lT-53"" 5-31-79 37-331 3^-1361 3-5-11-73 7-1721 5-7-347 3-4049 3-5-823 53-233 3-23-179 ■ 11-1123 5-7-353 32-1373 17-727 47-263 3-13-317 5-2473 83-149 - 3-7-19-31 89-139- 3^-13-103 17-709 5-2411 3-4019 31-389 7-1723 3-4021 5-19-127 11-1097 3^-149 29-419 3-4051 5-11-13-17 3^-'7-"l93 " " 3-7-593 5-47-53 3-5-19-43 7-17-103 13-23-41 3-61-67 3-4153 17-733 112-103 32-5-277 7-13-137 37-337 3-4157 3-5-811 23= 43-283 3-4057 7-37-47 5=-487 3^-11-41 19-641 13-937 3-31-131 5-2437 7-1741 3-17-239 73-167 89-137 3''-5-271 5-11-223 32-29-47 7-1753 3-4091 52-491 3-5''-7«23 13-929 47-257 3-4027 43-281 5-2417 3^-17-79 7-11-157 107-113 3-29-139 5-41-59 32-53-11 52-499 3-4159 3-4093 3-4127 7-29-61 5-2477 3-4129 13-953 7-1783 32-19-73 5-11-227 71.173 3»-5-7-13 11-1117 3-23-181 3-17-241 19-647 5-2459 3-4099 72-251 3"- 17 5-37-67 72-11-23 3-4133 13-31* 3-5-72-17 3-37-109 ii-iioo 29-431 OF ODD NUMBERS. 121 Prom 12,500 to 12,600 12,600 12,700 12,700 12,800 12,800 12,900 12,900 13,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 ,51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 32-463 5-4i-6i 3-ll'379 7-1787 13-977 3'-5-'7-'ii="" 97-131 71-179 3-19-223 3-17-251 7-31-59 5-13-197 3''-1423 23-557 3-4271 5-11-233 7-1831 3-4273 7-19-97 3-11-17-23 5-29-89 3-4201 5-2521 7-1801 33-467 3-13-331 3-43-97 5-2503 37-349 3=-5-7-41 3-5-29= 11-31-37 3-7-601 13-971 53-10] 3-23-61 73-173 17-743 3- 4211 5-7-19'' 5-2543 3"- 157 7-23-79 3-4241 5' -509 11-13-89 3-4243 29-439 7-17-107 3=-5-283 47-271 3^-13-107 19-659 7-1789 3-5^-167 11-17-67""' 3-4177 83-151 ' 5-23-109 3^-7-199 3-59-73 33-5«-19 101-127 5=-ll-47 3-31-139 7-1847 67-193 33-479 5-13-199 17-761 ■ 3-19-227 3-7-13-47 41-313 5-17-151 3-11-389 37-347 3-11-383 3-31-137 3-37-113 5-13-193 47-269 3''-5-281 '3= -1427 5-7-367 29-443 3-4283 71-181 . 7-43= 3-5-863 11=- 107 23-563 3=-1439 5-2549 3-7-607 11-19-61 41-311 3=-13-109 5-2551 3-47-89 7-11-163 7-13-139 3-4217 ^ 3^-5-31 29-433 19-661 3-53-79 17-739 5-7-359 3-59-71 13-967 3«-ll-127 5''- 503 5-2531 3-4219 3-5-857 13-23-43 7-11-167 3=-1429 19-677 5-31-83 3-4289 17-757 61-211 3-7-613 53-103 79-163 3= -53 11-1171 13-991 3-5-859 7=-263 5-2591 3-7-617 3-4253 7-1823 11-1151 33-7-67 5-17-149 53-239 3-41-103 13-997 3-29-149 5-2593 3-5-23-37 17-751 113= 3'- 11 -43 53.241 5=-7-73 3-4259 13-983 3=-ll-131 7-17-109 19-23-29 3-5=- 13= 7-1811 31-409 3' -1409 11-1153 5-43-59 3-4229 3-5=-173 , 19-683 3-7-599 23-547 3-4327 3-4261 5-2557 19-673 3»-7=-29 ii-'iies 3-5-853 67-191 3-5-839 41-307 5-73-53 33-13-37 31-419 11-1181 3-61-71 5-23-113 41-317 3-7-619 3" -1399 7^-257 5-11-229 3-13-17-19 43-293 73-37 3-4231 5-2539 3-4297 5-2579 3= -1433 3=-17-83 16 122 PRIME FACTORS From 13,000 to 13,100 1'3,100 13,200 13,200 13,300 13,300 13,400 13,400 13,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3^-5'17« 3-4337 7-ll'13' 5-19-137 3-4339 47-277 29-449 3^-1447 5' -521 7-1861 3-43- XOl 83-157 3'-5-n-79* is-ii'-si" 3*-7-23 5-2609' ' 3-4349 31-421 3-19-229 5-7-373 11-1187 3^-1451 37-353 3-5-13-67 73-179 7-1867 3-4357 17-769 5^-523 3'i-1453 11-29-41 103-127 3-7''-89 5-2617 23-569 3-4363 13-19-53 33-5-97 7-1871 3-11-397 5-2621 3-17-257 7-1873 3'-31-47 5-43-61 13-1099 3-4373 11-1193 3-5''-7 19-691 3^-1459 23-571 5-37-71 3-29-151 7-1877 17-773 3-13-337 5-11-239 3^-487 7-1879 3-5-877 59-223 3-41-107 5-2633 3'i-7-ll-19 13-1013 3-4391 5^-17-31 3-23-191 78-269 3^-5-293 iiMog'" 3-4397 79-167 5-7-13-29 3-53-83 67-197 43-307 3^-163 5-19-139 47-281 3-7-17-37 11-1201 73-181 3-5-881 3'-13-113 7-1889 5«-23« 3-4409 101-131 3-11-401 5-2647 7-31-61 3«-1471 17-19-41 3-5-883 13-1019 3-7-631 29-457 5-11-241 3»-491 89.149 3-4421 5-7-379 3-4423 23-577 13-1021 32- 5' -59 11-17-71 7' -271 3-19-233 37-359 5-2657 3-43-103 97-137 3«-7-211 5-2659 3'-ii-i3-3i* 47-283 53-251 3-5-887 7-1901 33-17-29 5-2663" 3-23-193 19-701 7-11-173 3-4441 5='-13-41 3^-1481 67-199 3-5-7-127 3-4447 11-1213 5-17-157 3«-1483 7-1907 13= -79 3-4451 5-2671 19=-37 3;61-73 31-431 7-23-83 3^-5-11 29-461 3-4457 43-311 53-107 3-73-13 17-787 3"- 1487 5-2677 11-1217 3-4463 7.1913 59-227 3-5-13-47 3^-1489 13-1031 5-7-383 3-41-109 11-23-53 3-17-263 5-2683 33-'7-'7l'" 31-433 3-5»-179 29-463 13-1033 3-1P-37 7-19-101 5-2687 3^-1493 89-151 3-4481 5-2689 7-17-113 3-4483 11-1223 33-5-13-23 43-313 3-7-641 5-2693 3-67^ 19-709 33-499 3-4493 ""' 13-17-61 97-139 3-5-29-31 3«-1499 103-131 5-2699 3-11-409 OP ODD NUMBERS. 123 From 13,500 to 13,600 13,600 13,700 13,700 13,800 13,800 13,900 13,900 14,000 . 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 23-587 3-7-643 5-37-73 13-1039 3''-19-79 59-229 3-5-17-53 7-1931 11-1329 3-4507 5^-54l" s^-ie? 83-163 7-1933 3-13-347 5-2707 3-4513 11-1231 29-467 3'i-5-7-43 19-23-31 17-797 3-4517 5-2711 3-4519 7-13-149 71-191 3^-11-137 5-2713 3-4523 41-331 7''-277 3-5^-181 37-367" 3^-503 17^-47 5-11-13- 3-7-647 107-127 19 3-23-197 5-2719 7-29-67 61-223 3-5-907 11-1237 31-439 3-13-349 5-7-389 32-17-89 53.257 3-19-239 5^-109 3-7-"lT-59' 43-317 3'-5-101 13-1049 23-593 3-4547 7-1949 5-2729 3-4549 11-17-73 3^-37-41 5-2731 7-1951 3-29-157 19-719 13-1051 3-5-911 79-173 3'-7'-31 112-113 52-547 3-47-97 3-4561 5-7-17-23 3^-13= 32-1511 3-S-11-83 7 -19 -103" 3-4567 71-193 5-2741 32-1523 3-7-653 5-13-211 11-29-43 3-17-269 32-52-6I 7-37-53 3-23-199 31-443 5-41-67 3-19-241 11-1249 7-13-151 3' -509 5-2749 59-233 3-4583 17-809 3-5-7-131 32-11-139 5-2753 3-13-353 72 -281 47-293 3-4591 52-19-29 23-599 32-1531 7-11-179 3-5-919 17-811 3-4597 13-1061 5-31-89 3'-7-73 37-373 3-43-107 5-11-251 3-4603 7-1973 19-727 32-5-307 41-337 13-1063 3-17-271 23-601 52-7-79 3-11-419 32-29-53 5-2767 101-137 3-7-659 109-127 3-5-13-71 61-227 11-1259 38-19 7-1979 5-17-163 3-31-149 83-167 3-4621 5-47-59 72-283 32-23-67 11-13-97 3-5='-37 3-7-661 5-2777" 32-1543 17-19-43 29-479 3-11-421 5-7-397 13-1069 3-41-113 3=-5-103 7-1987"' 3-4637 5-ii2'-23* 3-4639 31-449 32-7-13-17 52-557 19-733 3-4643 ^ 3-5-929 7-11.181 53-263 32-1549 73-191 5-2789 3-4649 13-29-37 7-1993 3-4651 5-2791 17-821 33-11-47 23-607 3-5-"72"-'i9' ■ 6i-229"" 3-4657 89-157 52-13-43 32-1553 7-1997 11-31-41 3-59-79 5-2797 71-197 3-4663 17-823 7-1999 32-5-311 124 PRIME FACTORS From 14,000 to 14,100 14,100 14,200 14,200 14,300 14,300 14,400 14,400 14,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-13-359 11-19-67 5-2801 3-7-23-29 59-239 3'- 1567 5-7-13-31 11-1291 7-2029 3-5-947 3"-7-227 3-4801 5-43-67 5-2861 3-19-251 41-349 11-1301 3-13-367 5-7-409 103-139 3"-37-43 3-4703 103-137 11-1283 3-5-941 19-743 7-2017 3'-523 29-487 53-113 3-17-277 71-199 13-1087 3-7-673 5-11-257 ■ 67-211 3^-1571 79-179 13-1093 3=-1579 61-233 5-2843 3-7-677 59-241 3-"li-43i""" 5". 569 41-347 3^-17-31 7-19-107 43-331 3-5-13-73 23-619 29-491 3-47-101 3"- 1601 3^-173 5-2803 107-131 3-4673 7-2003 37-379 3-5^-11-17 13' -83 7-29-71 3-5-31" 13-1109 3-11-19-23 3-5"- 191 3". 577 3"-7-229 47-307 7-23-89 3-17-281 11-1303 5-47-61 3= -59 13-1103 3= -1559 5- 7-461 3-4679 101-139 19-739 3-31-151 5-53" 11-1277 3"-7-223 i3-'23-'47 "" 3-5-937 17-827 3-43-109 73.41 5-29-97 3^-521 11-1279 3-17-283 5-2887 3-4813 7-2063 11-13-101 33.5-107 3-7-683 5-19-151 3-5-23-41 7-43-47 5-7-11-37 3^-1583 3-4783 113-127 31-463 3"-5-ll-29 7" -293 83-173 3-4787 53-271 5- 13'- 17 3-4789 7-2053 , 3'i-1597 5«-23 11-1307 3-4793 73-197 19-757 3-5-7-137 3-53-89 3-4817 97-149 5-7"-59 3-61-79 19-761 3-4751 5-2851 53-269 3-7"-97 13-1097 17-839 3"-5-317 11-1297 19-751 3-67-71 7-2039 5«-571 3-4759 109-131 5-19-149 3'-lP-13 7»-17« 3-4721 5-2833 31-457 3-4723 37-383 32-1607 , 5-11-263 17-23-37 3-7-13-53 29-499 41-353 3-5"-193 31-467 3-4691 5"- 563 7-2011 3-13-19' 3^-5"- 7 11-1289 3-29-163 13-1091 5-2837 3-4729 7-2027 23-617 3= -19-83 5-17-167 3"- 1609 7-2069 5-2897 3-11-439 3' -23" 5-2857 7-13-157 3-11-433 31-461 3"-5-313 73-193 3-7-11-61 17-829 5-9819 3-37-127 23-613 3'-13-41 37-389 5-2879 3-4799 7-ll"-17 43-337 3-48^1 5-13-223 7-19-109 3''- 179 3-5-953 17-29" 79-181 3-4733 OF ODD NUMBERS. 125 From 14,500 to 14,600 14,600 14,700 14,700 14,800 14,800 14,900 14,900 15,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 31 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 17-853 3-5-967 89-163 11-1319 3-7-691 23- 631- 5-2903 3^-1613 13-1117 3-47-103 5«-7-83 73-199 3-29-167 11-1321 s'-'s'-'iV-'ig" 7-3ii67"" 3-37-131 5-2909 3-13-373 33-72-11 5-41-71 3-23-211 3-5-971 7-2081 17-857 ' 3"- 1619 13-19-59 5='-ll-53 3-43-113 61-239 7-2083 3-4861 5-2917 29-503 3'- 1621 3-5-7-139 11-1327 13-1123 3-31-157 17-859 5-23-127 3^-541 7-2087 19-769 3-4871 5-37-79 47-311 3-11-443 7-2089 32-53-13 3-4877 5-2927 3-7-17-41 11* 32-1627 5-29-101 97-151 3-19-257 T"'- 13-23 s-'s-'gn" 107-137 3^-181 11-31-43 5-7-419 3-4889 17-863 3-67-73 52-587 13-1129 32-7-233 53-277 3-5-11-89 19-773 37-397 3-59-83 7-2099 5-2939 3=-23-71 61-241 3-132-29 5-17-173 7-11-191 3-4903 47-313 4i-359"" 3-7-701 52-'l9-3l" 3-4909 11-13-103 32-1637 5-7-421 3-17' 23-641 3-5-983 73-43" 32-11-149 5-13-227 3-4919 29-509 3-7-19-37 5-2953 33-547 11-17-79 3-52-197 7-2111 3-13-379 5-2957 32-31-53 23-643 7-2113 3-4931 5-11-269 192-41 . 113-131 32-5-7-47 13-17-67 59-251 3-4937 5-2963 3-11-449 7-29-73 3^-61 52-593 3-4943 7-13-163 3-5-23-43 37-401 11-19-71 32-17-97 5-2969 3-72-101 31-479 3-4951 5-2971 83-179 32-13-127 7-11-193 89-167 3-5-991 3-4957 107-139 53-7-17 33-19-29 23-647 3-112-41 5-13-229 3-7-709 53-281 32-5-331 3-4967 7-2129 •5-11-271 3-4969 17-877 13-31-37 32-1657 5-19-157 7-2131 3-4973 43-347 3i52'-'l99" 11-23-59 3'-7-79 109-137 5-29-103 3-13-383 67-223 3-17-293 5-72-61 32-11 -Ysi ' 19-787 3-5-997 7-2137 3-4987 13-1151 5-41-73 32-1663 3-4933 47-317 11-1361 3-7-23-31 52-599 17-881 3-4993 71-211 3*'-'5-37 7-2141 13-1153 3-19-263 11-29-47 5-2999 3-4999 53-283 126 PRIME FACTORS From 15,000 to 15,100 15,100 15,200 15,200 15,300 16,300 15,400 15,400 15,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 7-2143 3«-1667 5-3001 43-349 3-5003 17-883 3'-563 23-661 5-3041 3-37-137 67-227 7-41-53 3-11-461 5-17-179 11-13-107 3-5101 5-3061 3M 61-251 11-1373 3-5-19-53 29-'52i 3»-23-73 7-17-127 5-3023 3-5039 13-1163 73-211 3-5-13-79 7-31-71 19-811 3-11-467 ■3-5-7-11-13 3-5-1021 17».53 5-3083 3^-571 17-907 7-2203 3-53-97 5' -617 23-653 3«-1669 83-181 5^-601 3-5009 7-19-113 3«-19-89 31-491 13-1171 3-5«-7-29 3-5107 7-11-199 5»-613 3«-13-131 3- 19 -269"" 5-306? 7«-313 3-5113 23' -29 67-229 3«-5-ll-31 103-149 3-7P S>-1V 7-2161 32-4P 97-157 3-5077 3-37-139 13-1187 11-23-61 3'-5-7' 43-359 3-5011 5-31-97 11-1367 3^-557 13'- 89 3-5-17-59 41-367 101-149 3-29-173 37-409 3-5-1009 5-11-277 3=-1693 7«-311 3-7'-103 19-797 5-13-233 3^-11-17 3-5147 5-3089 3-19-271 7-2207 3-5081 5-3049 79-193 3-13-17-23 101-151 7-2179 3'-5-113 11-19-73 109-139 3-5051 5-7-433 23-659 3-31-163 3-7-17-43 13-1181 5-37-.83 3-5119 3'- 17- 101 5-11-281 13-29-41 3-5153 5-3011 32-7-239 11-37^ 3-5087 3-5021 5-23-131 13-19-61 3-5023 7-2153 59-257 3^-5-337 29-523 7-11-197 3-13-389 3^569 5-7-439 11'- 127 V 3-47-109 19-809 7.47s 3-5-1031 5-43-71 3-7-727 31-499 3*- 191 3^-1697 5^-13-47 3'-5=-67 5«-607 3-5059 43-353 17-19-47 3^-7-241 5-3037 3-6i-83 11-1381 3-5'-41 5'- 619 3-7-11-67 23-673 113-137 3-13-397 5-19-163 17-911 3'- 1721 7-2213 17-887 3-11-457 3-ii-463 7-37-59 17-29-31 3-5-1019 7-13^ 3' -1709 s-'n-isi"" 3-23-223 11-1399 5-7-431 3-47-107 79-191 3'- 1699 41-373 5-7-19-23 3-5099 3^-13-43 5-30ia 31-487 3-7-719 3-7-733 5-3079 89-173 3'-29-59 3-5-1013 7-13-167 3-5-1033 11-1409 OF ODD NUMBERS. 127 From 15,500 to 15,600 15,600 15,700 15,700 15,800 15,800 15,900 15,900 16,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-5167 37-419 5-7-443 3^-1723 13-1193 3-5171 5-29-107 59-263 3-7-739 11-17-83 19=- 43 3'-5'-23 53-293 3-31-167 7«-317 5-13-239 3-5179 41-379 32-11-157 5-3109 7-2221 3-71-73 103-151 3-5-17-61 47-331 3'-7-13-19 79-197 5-11-283 3-5189 23-677 3-29-179 5''-7-89 37-421 3^-577 3-5-1039 11-13-109 7-17-131 3-5197 31-503 5-3119 3= -1733 19-821 3-7-743 5-3121 3-1P-43 67-233 13-1201 3^-5-347 7-23-97 . 3-41-127 17-919 3-5209 7^-11-29 3^-193 5-53-59 19-823 3-13-401 3-5-7-149 3'-37-47 11-1423 5-31-101 3-17-307 7-2237 . 3-23-227 5-13-241 3= -1741 7-2239 S-S^'-U-W 61-257 3-5227 5-3137 3»-7-83 29-541 13-17-71 3-5231 5-43-73 11-1427 3-5233 7-2243 41-383 3»-5-349 113-139 23-683 3-5237 19-827 5-7-449 3-13'-31 11-1429 79-199 3»-1747 52-17-37 3.7=-107 3-5-1049 3'-ll-53 7-13-173 5-47-67 3-29-181 19-829 3-59-89 5-23-137 7-2251 32-17-103 11-1433 3-5-1051. 13-1213 3-7-751 S'-'oVl" 32-1753 31-509 43-367 3-5261 5-7-11-41 3-19-277 17-929 3*-5-13 7-'37i6i' 3-23-229 5-29-109 3-11-479 97-163 32-7-251 5-3163 3-5273 13-1217 3-52-211 72-17-19 11-1439 32-1759 71-223 5-3167 3-5279 47-337 7-31-73 3-5281 5-3169 13-23-53 3^-587 112- 131' 83-191 3-5-7-151 101-157 3-17-311 29-547 5-19-167 32-41-43 7-2267 59-269 3-11-13-37 5^-127 3-67-79 7-2269 3=-5-353 3-5297 23-691 5-11-172 3-7-757 13-1223 33-19-31 5-3181 3-5363 " 7-22-73 h'-h'-mi 11-1447 32-29-61 52-72-13 3-5309 17-937 89-179 3-47-113 5-3187 32-7-'ll-'23 ' 19-839 107-149 3-5-1063 37-431 41-389 3-13-409 7-43-53 5-3191 3^-197 11-1451 3-17-313 5-31-103 7-2281 3-5323 32-52-71 13-1229 19-292 3-7-761 11-1453 5-23-139 3-732 59-271 32-1777 5-7-457 17-941 3-5333 128 PRIME FACTORS From 16,000 to 16,100 16,100 16,200 16,200 16,300 16,300 16,400 16,400 16,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 13-1231 3-5-11-97 7-2287 "" 3' -593 67-239 5-3203 3-19-281 83-193 37-433 3-7=^-109 5^-641 11-31-47 3' -13 -137 17-23-41 3-5-1069 7-29-79 43-373 3-5347 61-263 5-3209 3^-1783 11-1459 7-2293 3-5351 5-13^-19 5^-643 3-23-233 7-2297 13-1237 3'- 1787 5-3217 3-31-173 7-1P-19 3-5-29-37 3^^-1789 5-3221' "" 5-7-13-59 89-181 3-41-131 5-11-293 71-227 3^-199 73.47 23-701 3-5^-43 127« 3-19-283 13-17-73 5-7-461 3^-11-163 3- 53 101 3? -5-7-17 3-11 487 3-5381 5-3229 67-241 3-7-769 31-521 29-557 3^-5-359 107-151 11-13-113 3-5387 7-2309 5-53-61 3-17-317 19-23-37 103-157 3^-599 5^-647 7-2311 3-5393 11-1471 3-5-13-83' 3'-7-257 17-947 5-41-79 3-5399 I 97-167 17-953 3-11-491 5-7-463 19-853 3«-1801 13-29-43 31-523 3-5-23-47 7^-331 3-5407 5»-'l'l'-'59 ' 3= -601 3-7-773 5-17-191 13-1249 3-5413 109-149 37-439 32-5-19" 7-11-211 3-5417 5-3251 3-5419 71-229 7-23-101 3'i-13-139 5-3253 3-11-17-29 53-307 3-5'-7-31 41-397 73-223 3= -67 19-857 5-3257 3-61-89 7-13-179 11-1481 3-5431 5-3259 43-379 3i'-1811 7-17-137 3-5-1087 23-709 47-347 3-5437 11-1483 5-13-251 3«-7'-37 19-859 3-5441 5«-653 29-563 3-5443 7.2333 3'-5-ll« 17-31* 3-13-419 59-277 5-7-467 3-5449 83-197 3«-23-79 5-3271 11-1487 3-7-19-41 3-5-1091 13-1259 3»-17-107 7-2339 5'- 131 3-53-103 11-1489 3-43-127 5-29-113 7-2341 3»'607 37-443 13'-97 3-5-1093 19-863 23«-31 3-7-11-71 47-349 5-17-193 3«-1823 61-269 3-5471 5'7«-67 3'-i3-'42i' 11-1493 3«-5»'73 7-2347'" 3-5477 5-19-173 3-5479 17-967 41-401 3<-7-29 5-11-13-23 3-5483 3-5-1097 7-2351 109-151 3*-31-59 101-163 5-37-89 3-11-499 43-383 7-13-181 3-172-19 5«-659 s^-'is'si 53-311 3-5-7-157 ii-*i499"' 3-23-239 5'-3299" '• 3»-13-47 7-2357 OP ODD NUMBERS. 129 From 16,500 to 16,600 16,600 16,700 16,700 16,800 16,800 16,900 16,900 17,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 as 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 29-569 3-5501 5-3301 17-971 3-5503 11-19-79 7=-337 3^-5-367 83-199 3-5507 13-31-41 5«-661 3-7-787 61-271 3«-ll-167 3-3307 23-719 3-37-149 7-17-139 71-233 3-5-1103 is-ig-'e?" 3'- 613 5-7-'ii'-43' 3-5519 29-571 3-5521 5-3313 3^-7-263 73-227 3-5''-13- IP- 137 59-281 17 3-5527 7-23-103 5-31-107 3«-19-97 53-313 47-353 3-5531 5-3319 7-2371 3-11-503 It 13-1277 17-977 3-7^-113 37-449 5-3323 3-29-191 11-1511 3''-1847 53-7-19 13-1279 3-23-241 3-5-1109 127-131 7-2377 3'-43= 11-17-89 5-3329 3-31-179 3-7-13-61 5-3331 3'- 617 19-877 3-5-11-101 7-2381 79-211 3-5557 5'-23-29 3^-17-109 13-1283 7-2383 3-67-83 5-47-71 11-37-41 3-5563 3»-5-7-53 59-283 3-19-293 5-13-257 3-5569 7=^-11-31 17-983 33-619 5-3343 73-229 3-5573 23-727 7-2389 3-5^-223 43-389 3^-11-13' 29-577 5-3347 3-7-797 19-881 3-5581 5-17-197 3^-1861 7-2393 11-1523 3-5-1117 13-1289 3-37-151 5-7-479 315-23 41-409 31-541 3-5591 5^-11-61 19-883 3-7-17-47 97-173 13-1291 3^-5-373 103- 163 3-29-193 7-2399 5-3359 3-11-509 107-157 53-317 3^-1867 5-3361 7^ 3-13-431 17-23-43 3-5-19-59 67-251 IP -139 33-7-89 5^-673 3-71-79 3-31-181 5-7-13-37 113-149 3^-1871 11-1531 3-5-1123 17-993 7«29-83 3-41-137 19-887 5-3371 3''-1873 23-733 13-1297 3-7-11-73 5-3373 101-167 3-5623 47-359 33-5" 7-2411 3-17-331 5-11-307 3-13-433 7-19-127 3''-1877 5-31-109' 61-277 3-43-131 3-5-7^-23 11-29-53 37-457 3^-1879 13-1301 5-17-199 3-5639 7-2417 3-5641 5= -677 3"- 11 -19 7-41-59 3-5-1129 13-1303"" 3-5647 5-3389 3='-7-269 17-997 11-23-67 3-5651 5-3391 31-547 3-5653 7-2423 3^-'5'-"l3-"2"9*' 19^-47 71-239 3-5657 11-1543 5=-7-97 3-5659 33-17-37 5-43-79 3-7-809 13-1307 3-5-11-103 23-739 89-191 130 PRIME FACTORS Prom 17,000 to 17,100 17,100 17,200 17,200 17,300 17,300 17,400 17,400 17,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3= -1889 5-19-179 3-5669 73-233 3-53-107 5-41-83 7-11-13-17 3=^-31-61 29-587 3-5''-227 3-7-811 5-3407 3^-631 11-1549 3-13-19-23 5-7-487 3-5683 17^-59 3^-5-379 37-461 7-2437 3-11^-47 113-151 5-3413 3-5689 13^-101 43-397 3''-7-271 5^-683 3-5693 19-29-31 11-1553 3-5-17-67 7-2441 23-743 3" -211 5-13-263" 3-41-139 7=-349 3-5701 5-11-311 3'- 1901 71-241 109-157 3-5-7-163 17-19-53 3-13-439 5^-137 3'-ll-173 7-2447 37-463 3-5711 5-23-149 3-29-197 61-281 7-31-79 33-5-127 13-1319 11-1559 3-5717 17-1009 5-47-73 3-7-19-43 13P 3'' -1907 5-3433 3-59-97 . 7-11-223 13-1321 3-5=-229 89-193 41-419 3«-23-83 5-7-491 3-17-337 3-11-521 5-19-181 29-593 33-7«-13 103-167 3-5-3i-37' 3-5737 7-2459 5-11-313 3^-1913 67-257 17-1013 3-5741 5=-13-53 7-23-107 3-5743 19-907 3»-5-383 11-1567 3-7-821 43-401 5-3449 3-5749 47-367 13-1327 3^-71 5-7-17-29 3-11-523 41-421 61-283 3-5-1151 31-557 7-2467 3"- 19 -101 23-751 5'- 691 3-13-443 37-467 11-1571 3-7-823 5-3457 59-293 3»-17-113 3-5-1153 7'i-353 3-73-79 ll'-13 5-3461 3^-641 19-911 7-2473 3-29-199 5-3463 3-23-251 17-1019 3»-5='-7-ll is-ai'-'is" 3-53-109 5-3467 3-5779 7-2477 3'- 41-47 5-3469 11-19-83 3-5783 7-37-67 3-5-13-89 17-1021 3^-643 97-179 5-23-151 3-7-827 11-1579 29-599 3-5791 5^-139 3'-1931 7-13-191 s-'s-'ig-ei" 3-11-17-31 5-v-n"" 3= -1933 127-137 3-5801 5-59» 13«-103 3-7-829 23-757 11-1583 3<-5-43 3-5807 7-19-131 5=- 17-41 3-37-157 29-601 3»-13-149 5-11-317 7-47-53 3-5813 107-163 s-'s-'iies" ' 73-239 32.7.277 31-563 5-3491 3-11-23' 13-17-79 19-919 3-5821 5-7-499 3'-"64y"' 101-173 3-5»-233 7-11-227 3-5827 5-13-269 3'-29-67 3-73-17 5-3499 3-19-307 OP ODD NUMBERS. 131 From 17,500- to 17,600 17,600 17,700 17,700 17,800 17,800 17,900 17,900 18,000 1 3 S 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 ll-37'43 23-761 3'-5-309 7-41-61 3-13-449 83-211 5-31-113 3-5839 7-2503 3^-11-59 5=-701 17-1031 3-5843 47-373 89-197 3-5-7-167 13-19-71 3'- 1949 53-331 5-11^-29 3-5849 7-23-109 3-5851 5-3511 97-181 3^-1951 17-1033 7-13-193 3-5-1171 11-1597 3-5857 5»-19-37 3«-7-31 3-5867 29-607 5-7-503 3-5869 3-5861 5-3517 43-409 3-11-13-41 7»-359 73-241 3''-5-17-23 11-1601 3'' -19 -103 5-13-271 79-223 3-7-839 67-263 3-5'-47 i7''-61 3' -653 11-7-229 5-3527 3-5879 31-569 13-23-59 3-5881 5-3529 7-2521 3'^ -37 -53 19-929 127-139 3-5-11-107 3-7-29' 17-1039 5-3533 3^-13-151 41-431 3-43-137 5^-7-101 11-1607 3-71-83 33-5-131 23-769 72.19! 3-5897 13-1361 5-3539 3-17-347 11-1609 31-571 3'-7-281 5-3541 3-5903 89-199 3-5-1181 7-2531 13-29-47 3^-11-179 37-479 5^-709 3-19-311 7-17-149 3-23-257 5-3547 3^-73 113-157 11-1613 3-5-7-13= 3-61-97 41-433 5-53-67 32-1973 7-43.59 3-31-191 5-11-17-19 109-163 3-5923 13-1367 7-2539 3«-5''-79 29-613 23-773 3-5927 5-3557 3' -659 5-3559 13 -37''' 3-17-349 7-2543 19-937 3-5-1187 11-1619 3"- 1979 47-379 5-7-509 3-5939 103-173 71-251 3-13-457 5=-23-31 3^-7-283" 11-1621 17-1049 3-5-29-41 3-19-313 7-2549 5-43-83 33 -661 13-1373 3-11-541 5-3571 7-2551 3-5953 53-337 3=^-5-397 17-1051 107-167 3-7-23-37 61-293 53-11-13 3-59-101 19-941 3=^-1987 5-7''-73 31-577 3-67-89 29-617 3-5-1193 11-1627 7-2557 3^-13-17 5-3581 '" 3-47-127 3-7-853 5-3583 19-23-41 3'-ll-181 3-52-239 7-13-197 3-43-139 79-227 5-17-211 3'- 1993 7-11-233 3-5981 5-37-97 131-137 3-31-193 29-619 13-1381 33-5-7-19 3-5987 11-23-71 5-3593 3-53-113 7-17-151 3'- 1997 5''-719 3-'i3-46i" 7^-367 3-5-11-109 3^-1999 19-947 5-59-61 3-7-857 41-439 132 PRIME FACTORS From 18,000 to 18,100 18,100 18,200 18,260 18,300 18,300 18,400 18,400 18,500 . 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 47-383 3-17-353 5-13-277 11-1637 3^-23-29 7-31-83 23-787 43-421 3-5-17-71 19-953 7-13-199 3-6037 59-307 5-3623 33'11-61 3-6067 109-167 5-11-331 32.7.172 131-139 3-13-467"" 5-3643 3-6101 5-7-523 7-11-239 3''-5-409 79-233 41-449 3-17-19» 3-17-359 3-5-1201 43-419 37-487 3-6007 67-269 5'- 7- 103 3^-2003 IF- 149 13-19-73 3-eoii 5-3607 17-1061 3-7-859 3^-5-11-37 13-1409 7-2617 3-31-197 73-251 5= -733 3-41-149 5-29-127 3-7-877 113-163 13»-109 3^-23-89 5«-ll-67 3-6073 7-19-137. 3»-"52 11-1657 3-7-863 5^-29 3-6043 3-6143 7-2633 . 3-59-103 23-797 33-7-97 5-19-193 11-1667 3-6113 3=-5-13-31 7-2591 11-17-97 3-6047 5-7-521 3-6079 13-23-61 17-29-37 3«-2027 5-41-89 71-257 3-7-11-79 3-5-1229 103-179 3'- 683 5-'7-i7-3i"' 3-11-13-43 19-971 13-17-83 3-5-1223 7-2621 59-311 3«-2039 3''-5-401 3-11-547 7-2579 5-23-157 3-13-463 5-19-191 3-23-263 7-2593 3'-2017 5-3631 67-271 3-6053 11-13-127 41-443 3-5-7-173 37-491 3-6151 5-3691 3- 5- 1217 i9-3F 3«-2029 7-2609 5-13-281 3-6089 5-3671 3-29-211 11-1669 7-43-61 3-6121 5-3673 3'-7-293 3^-223 5-3613 7-29-89 3-19-317 17-1063 11-31-53 3-5^-241 ibi-ira 3j.72.41 13^-107 5-3617 3-6029 37-499 3-5-1231 59-313 11-23-73 3-47-131' 72-13-29 5«-739 3«-2053 17-1087 3^-13-157 3»-673 17-1069 5"- 727 3-73-83 7'-53 IP- 151 3-6091 5'-17-43 7'' -373 3' -677 101-181 47-389 3-5-23-53 19-967 3-53-7« 17"23-47 3-11-557 31-593 5-3677 3*- 227 7-37-71 53-347 3-6131 5-13-283 3-11-19-29 5-3637 13-1399 3'-43-47 7-23-n3"'" 3-5-1213 31-587 3-61-101 5-3697 7-19-139 3-6163 11-41' 3-7-13-67 11-1663 5-3659 3'i-19-107 29-631 79-229 3-37-163 5-7-11-47 3'-5-137 53-349 13-1423 3''-2Dll 3-6133 ... OP ODD NUMBERS. 133 From 18,500 to 18,600 18,600 18,700 18,700 18,800 18,800 18,900 18,900 19,000 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 3-7-881 5-3701 3-31'199 83-293 107-173 3'-ll=-17 5-7-23' 3-6173 3-5''-13-19 97-191 7-2647 3=-29-71 43-431 5-11-337 .3-37-167 3-7-883 5-3709 17-1091 3^-229 13-1427 3-5-1237 7-11-241 67-277 3-23-269 19-977 5-47-79 3" -2063 31-599 7=^-379 3-41-151 52-743 13-1429 3-11-563 81 17-1093 83 85 3^-5-7-59 87 89 91 93 95 97 99 29-641 3-6197 5-3719 3-6199 7-2657 11-19-89 3'-13-53 5-61' 23-809 3-6203 37-503 7-2659 3-5-17-73 3-2089 43-433 3'-2069 11-1693 53-149 3-7-887 13-1433 31-601 3-6211 5-3727 3' -19 -109 7-2663 103-181 3-5-11-113 29-643 17-1097 3-6217 23-811 5-7-13-41 33-691 47-397 3-6221 5-3733 11-1697 3-7'-127 71-263 3'-5'-83 19-983 3-13-479 7-17-157 5-37-101 3-6229 11-1699 3'-31-67 5-3739 7-2671 3-23-271 59-317 3-5-29-43 13-1439 53.353 3^-7-11 5-19-197 3-17-367 97-193 3-79= 5'-7-107 61-307 3'-2081 11-13-131 3-5-1249 41-457 7-2677 3-6247 5-23-163" ' 3'-2083 17-1103 3-7-19-47 5-1P-31 3-13'-37 73-257 29-647 33-5-139 7' -383 137' 3-6257 5'- 751 3-11-569 89-211 7-2683 3'-2087 5-13-17' 3-6263 19-23-43 3'-5-'7-"i79' n'-im" 5-3761 3-6269 7-2687 13-1447 3-6271 5-53-71 31-607 33-17-41 11-29-59 7-2689 3-5'-251 67-281 19-991 3-6277 37-509 5-3767 3'-7-13-23 83-227 3-11-571 5-3769 47.401 3-61-103 7-2693 17-1109 3'- 5- 419 109-173 3-6287 13-1451 5-73-11 3-19-331 113-167 3<-233 53-151 43-439 3-7-29-31 79-239 23-821 3-5-1259 11-17-101 13-1453 3'-2099 7-2699 5-3779 3-6299 41-461 3-6301 5-19-199 7-37-73 3'-ll-191 3-5-13-97 3-7-17-53 127-149 5'- 757 33-701 23-823 11-1721 3-6311 5-7-541 29-653 3-59-107 13-31-47 19-997 3'-5-421 7-2767 3-6317 11-1723 5-17-223 3-T1-89 67-283, 3'-7'-43 5-3793 13-1459 3-6323 61-311 3-5'-ll-23 7-2711 33-19-37 41-463 5-3797 3-6329 17-1117 7-2713 3-13-487 5-29-131 11'- 157 3'-2111 134 PRIME FACTORS From 19,000 to 19,100 19,100 19,200 19,200 19,300 19,300 19,400 19,400 19,500 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 1 51 53 55 57 59 61 63 65 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3-6367 7-2729 5-3821 3''-ll-193 97-197 29-659 3-23-277 5-3823 7-2731 3-6373 13-1471 3=-53-17 31-617 11-37-47 3-7-9U 19»-53 5-43-89 3-6379 7-13-211 3-37-173 5-23-167 97-i99 3''-5-lM3 43-449 3-29-223 31-613 3-5-7-181 83-229 5-3881 3-6469 13-1493 7-47-59 3'- 719 5-11-353 3-19-337 3^-'s-V-6i 11-1747 3-6337 5- 8303 3^-2113 7-11-13-19 23-827 3-17-373 5= -761 53-359 3-6343 7-27i9 3^-5-47 3-41-157 7-31-89 5-3863 3-47-137 3-6473 3-43-149 47-409 5»-769 3-13-17-29 7-41-67 139' 3''-19-113 5= -773 7-11-251 3-17-379 13-1487 3-5»-7-37 3«-17-127 3»-2137 5-3847 3-5-1289 61-317 83-233 3''-7-307 23-29= 5-53-73 3-6449 11-1759 37-523 3-6451 5-7«-79 13-1489 3^-239 19-1019 17^-67 3-5-1291 107-181 7-2767 3-11-587 5-13S23 3-11-19-31 7-2777 79-241 3-11-577 137-139 5-13-293 3-7-907 43-443 3-1P-53 71-271 7-2749 3-5-1283 19-1013 3'- 709 5-T-547 41-467 3-13-491 11-1741 107-179 3-5-1277 7^-17-23""' 3«-2129 3-6481 5-3889 3«-2161 53-367 7^^-397 3-5-1297 3^-23-31 13-1481 5-3851 3-7^-131 3^-29-73 5-37-103 17-19-59 3-6353 7»-389 11-1733 3-5-31-41 23-829 11-29-61 3-13-499 U-17-103 3-6421 5-3853 5-3833 3-6389 29-661 19-1009 3-7-11-83 5i'.13-59 127-151 3^-2131 5-17-229 3''-7-103 3'-2141 7-2753 3=-13-163 s'-V-'iog"" 3-6359 3-6491 5»-19-41 3-5«-257 37-521 13-1483 3-6427 11-1753 5-7-19-29 3''-2143 101-191 3-59-109 5-17-227 23-839 3-7-919 5*- 31 3«-2153 3-43-151 7-ll«-23 3«-'5-433"" 13-1499 3-6361 5-11-347 3-7-13-71 5-3877 3-5-1279 7-2741 31-619 3-6397 17-1129 5-11-349 3° -79 73-263 3'-7-101 17-1123 61-313 3-5-19-67 13"- 113 71-269 3-23-281 ii-4i-43"*" 3»-5-431 7-17-163 19-1021 3-73-89 101-193 5-7-557 3-67-97 17-31-37 OF ODD NUMBERS. 135 Prom 19,500 to 19,600 19,600 19,700 19,700 19,800 19,800 19,900 19,900 20,000 1 3 5 7 9 11 13 IS 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 S3 SS 57 59 61 63 6S 67 69 71 73 75 77 79 81 83 85 87 89 91 93 95 97 99 3'- 11-197 5-47-83 3-7-929 109-179 13-19-79 3-5-1301 29-673 131-149 3^-241 7-2789 5''-n-71 3-23-283 59-331 3-1,7-383 5-3907 7-2791 3=-13-167 3-5-1303 11-1777 113-173 3-7'-19 5-3911 3«-41-53 31-631 3-6521 5-7-13-43 17-1151 3-11-593 23«-37 3'-5'-29 7-2797 3-61-107 5-3917 3-6529 19-1031 11-13-137 3^-7-311 5-3919 17-1153 3-5-1307 7-2801 3'-ll-199 17-19-61 5-7-563 3-6569 3«-2179 11-1783 5-3923 3-13-503 23-853 7-2803 3-31-211 5«-157 19-1033 3' -727 67-293 29-677 3-5-7-11-17 73-269 41-479 3-6547 13-1511 5-3929 3«-37-59 7'-401 43-457 3-6551 5-3931 11-1787 3-6553 7-53^ 3'-5-19-23 71-277 13-17-89 3-79-83 103-191 5^-787 3-7-937 11-1789 3' 5-31-127 3-6563 7-29-97 47-419 3-5-13-101 23-857 3-6571 5-3943 3^-'7-'313 " 13-37-41 ii'^-ies 3-5'-263 loa-isi 3-6577 7-2819 5-3947 3^-17-43 19-1039 3-6581 5-11-359 7''-13-31 3-29-227 3'-5-439 23-859 3-7-941 5-59-67 3-11-599 53.373 17-1163 S'^-IS' 52-7-113 3-47-139 3-19-347 131-151 73-271 3-5-1319 47*4^1 7-11-257 3' -733 5- 37 -107* 3-6599 13-1523 3-7-23-41 5-17-233 29-683 3^-31-71 11-1801 3-5-1321 7-19-149 3-6607 43-461 5«-13-61 3= -2203 79-251 7-2833 3-11-601 5-3967 83-239 3-17-389 3*- 5- 7^ 89-223 23-863 3-13-509 5-11-19^! 3-6619 7-2837 3^-2207 5-29-137 3-37-179 31-641 7-17-167 3-53-53 11-13-139 103-193 32.472 59-337 5-41-97 3-7-947 3-19-349 5-23-173 101-197 3'-ll-67 7-2843 13-1531 3-5-1327 17-1171 43-463 3-6637 5-7-569 3^-2213 11-1811 3-29-229 52-797 3-7-i3-73* 19-1049 31-643 32-5-443 127 -'157' 3-172-23 72-11-37 5-3989 3-61-109 71-281 3'- 739 5-13-307 7-2851 3-6653 3»5-lF 41-487 19-1051 32-7-317 52-'l7-47" 3-6659 13-29-53 3-6661 5-7-571 11-23-79 32-2221 3-5-31-43 7'-2857"'" TABLE OF THE PEIME FACTORS OF EVEN NUMBEES, fKOM 1 TO 12,500. 138 PRIME FACTORS From 100 200 300 400 to 100 200 300 400 500 2 2>3-17 2U0i 2-151 2-3-67 4 2= 2'- 13 22-3-17 2*- 19 22.101 6 2-3 2-53 2-103 2-32-17 2-7-29 8 2' 22.3' 2<-13 22-7-11 23-3-17 10 2-5 2-5'll 2-3-5-7 2-5-31 2-5-41 12 2«-3 2*-7 22.53 23-3-13 . 22-103 14 2-7 2-3'19 2-107 2-157 2-32-23 16 2" 22-29 23 -33 2'!- 79 25-13 18 2-3'> 2-59 2-109 2-3.53 2-11-19 20 2'-5 2».3-5 22.5.11 2».5 2«-3-5.7 22 2-11 2-61 2.3-37 2.7-23 2.211 24 2'-3 2=-31 2^-7 22-3* 23-53 26 2-13 2.3=-7 2-113 2-163 2-3-71 28 2=-7 2' 22-3-19 23-41 22-107 30 2-3'5 2-5-13 2-5-23 2.3.5-11 2-5.43 32 25 2'-3-ll 23-29 22.83 2^.33 34 2-17 2-67 I • 2.32-13 2.167 2.7-31 36 22.32 2= -17 22-59 2*.3-7 22.109 38 2-19 2-3-23 2-7-17 2-132 2.3.73 40 2»-5 2''-5-7 2«-3-5 32-5.17 2= .5. 11 42 2>3-7 2-71 2-112 2.32-19 2.13.17 44 2^-11 2''-3'' 22-61 23-43 22.3.37 46 2-23 2-73 2-3-41 2-173 2-223 48 2^-3 2'- 37 23.31 22-3-29 2«-7 50 2-52 2-3-5» 2.53 2-52.7 2.32.52 52 25i-13 2'- 19 22.32.7 25-11 2«.113 54 2-3' 2-7-11 2-127 2-3-59 2.227 56 22-7 2=i-3-13 28 22-89 23-3-19 58 2-29 2-79 2-3-43 2-179 2-229 60 2'i-3-5 »-5 22-5-13 23-32.5 22.5.23 62 2-31 2-3* 2-131 2-181 2.3.7.11 64 -2« 22-41 23-3-11 22-7-13 2*. 29 66 2-3-11 2-83 2-7-19 2-3-61 2.233 68 2^-17 23-3-7 22-67 2" -23 22.32-13 70 2.5-7 2-5-17 2-33-5 2-5-37 2-5-47 72 2»-3« 22-43 2^-17 22-3-31 23-59 74 2-37 2-3-29 2-m 22^-23 2-11-17 2-3-79 76 2" -19 2^-11 23-47 22-7-17 78 2-3-13 2-89 2-139 2-33-7 2-239 80 2'-5 2^-32-5 23-5-7 22-5-19 25-3-5 82 2.41 2-7-13 2-3-47 2-191 2-241 84 2'-3-7 23-23 22-71 2''-3 22-112 86 2-43 2-3-31 2-11-13 2-193 2.35 88 23-11 22.47 2^-32 22.97 23.61 90 2-3'-5 2-5-19 2-5-29 2.3.5.13 2.5.72 92 2''23 2«-3 22-73 23.72' 22-3-41 94 2-47 2-97 2-3-72 2-197 2-13-19 96 2^3 22.72 23-37 22.32-11 2* -31 98 2-7« 2-32-11 2-149 2-199 2-3-83 100 2' ■5'' 23-52 22-3-52 2«-52 • 22.53 OP EVEN NUMBERS. 139 From 500 600 700 800 900 to 600 700 800 900 1,000 2 2-251 2-7-43 2-33-13 9-401 2-11-41 4 23-3»-7 2'- 151 26- U 92-3-67 23-113 6 2-11-23 2-3-101 2-353 2-13-31 2-3-151 8 2»-127 2^-19 2«-3-59 93-101 22-227 10 2-3-5-17 2-5-61 2-5-71 2-31-5 2-5-7-13 12 2» 22-3^-17 23-89 22-7-29 21-3-19 14 9-257 2-307 2-3-7-17 2-11-37 2-457 16 2'-3-43 2^-7-11 2^-179 gi-s-n 22-829 18 2-7-37 2-3-103 2-359 2-409 2-33-17 20 2^-5-13 2=-5-31 2«-3'-5 22-5-41 23-5-23 22 2-3'-29 2-311 2-192 2-3-137 2-461 24 2'- 13] 2'-3-13 2'- 181 23-103 22-3-7-11 2f, 2-263 2-313 2-3- IP 2"- 7 -59 2-463 28 2<-3-ll 2^-157 23-7-13 22-32-23 25-29 30 2-5-53 2-3«-5-7 2-5-73 • 2-5-83 9-3-5-31 32 2'-7-19 2'- 79 2=-3-61 23-13 92-233 34 2-3-89 2-317' 9-367 2-3-139 2-467 36 2^-67 2=-3-53 9° -23 92-11.19 23-32-13 38 2-269 2-11-29 2-3=^-41 9-419 2-7-67 40 2'-3=-5 2'- 5 22.5-37 93-3-5-7 22-5-47 42 2-271 2-3-107 2-7-53 2-421 2-3-157 44 2^-17 2^-7-23 23-3-31 22-211 21-59 46 2-3-7-13 2-17-19 2-373 9-32-47 2-11-43 48 2»-137 23-3* 22-11-17 ^1-53 92-3-79 50 2- 5'^- 11 2-5^-13 2-3-53 9-52-17 9-52-19 52 2^-3-23 22-163 9-' -47 22-3-71 23-7-17 54 2-277 2-3-109 9-13-29 2-7-61 2-32-53 56 2- -139 2"- 41 92-33-7 23-107 92-239 58 2-3=-31 2-7-47 2-379 9-3-11-13 2-479 60 2^-5-7 2^-3-5-11 23-5-19 22-5-43 23-3-5 62 2-281 2-331 2-3-127 2-431 2-13-37 64 2''-3-47 2' -83 2^-191 95.33 22-941 66 2-283 2-3^-37 2-383 2.433 2-3-7-23 68 2'- 71 2^-167 28-3 92.7-31 23-112 70 2-3-5-19 2-5-67 2-^-7-11 2-3-5-99 ' 2-5-97 72 2^-11-13 2^-3-7 2^-193 93-109 92.35 74 2-7-41 2-337 2-32-43 9-19-93 2-487 76 26.32 2=-13« 93-97 92-3-73 21-61 78 2-17* 2-3-113 2-389 9-439 2-3-163 80 2«-5-29 23-5-17 22-3-5-13 9*-5-ll 92-5-72 82 2-3-97 9-11-31 2-17-23 2-32-72 2-491 84 2' -73 22-3=- 19 21.72 22-13-17 93-3-41 86 2-293 2-73 2-3-131 2-443 9-17-29 88 22.3.72 21-43 22-197 23-3-37 22-13-19 90 2-5-59 2-3-5-93 9-5-79 2-5-89 2-32-5-11 92 2^-37 2'- 173 93-32-11 22-293 25-31 94 2-3^-11 9-347 2-397 2-3-149 2-7-71 96 2^-149 23-3-29 22-199 9^-7 92-3-83 98 2-13-23 2-349 2-3-7-19 2-449 2-499 100 2^-3-52 2?-5^-7 2^-52 92.32-5^ 23-53 140 PRIME FACTORS From 1,000 1,100 1,200 1,300. 1,400 to 1,100 1,200 1,300 1,400 1,500 2 2-3-167 2-19-29 2-601 2-3-7-31 2-701 4 2»-251 2<-3-23 22-7-43 23-163 22-33-13 6 2-503 2-7-79 2-32-67 2-653 2-19-37 8 2*-3»-7 22-277 23-151 22-3-109 2'- 11 10 2-5-101 2-3-5-37 2-S-II2 2-5-131 ^ 2-3-5-47 12 2»-n-23 23-139 22-3-101 25-41 22-353 14 2-3-13' 2-557 2-607 2-32-73 2-7-101 16 2'- 127 22-32-31 2«-19 22-7-47 23-3-59 18 2-509 2-13-43 2-3-7-29 2-659 2-709 20 2'-3-5-17 2^-5-7 22-5-61 • 23-3-5-11 22-5-71 22 2-7-73 2-3-11-17 2-13-47 2-661 2-32-79 24 2'" 22-281 23-32-17 22-331 21-89 26 2-3'- 19 2-563 2-613 2-3-13-17 2-23-31 28 2=-257 23-3-47 22-307 2" -83 22-3-7-17 30 2-5-103 2-5-113 2-3-5-41 2-5-7-19 2-5-11-13 32 2=-3-43 22-283 2^-7-11 22-32-37 23-179 34 2-11-47 2-3^-7 2-617 2-23-29 2-'3-239 36 2'-7-37 2^-71 22-3-103 23-167 22-359 38 2-3-173 2-569 2-619 2-3-223 2-719 ■ 40 2'-5-13 . 22-3-5-19 23-5-31 22-5-67 35.32-5 42 2-521 2-571 2-33-23 2-11-61 2-7-103 44 2'-3«-29 23-11-13 22-311 2e-3-7 22-192 46 2-523 2-3-191 2-7-89 2-673 2-3-241 48 2^-131 22-7-41 2*-3-13 22-337 23-181 50 2-3-52-7 2-52-23 2-5< 2-33-52 2-52-29 52 22-263 2'- 32 22-313 23-132 22.3-112 54 2-17-31 2-577 2-3-11-19 2-677 2-727 . 56 25-3-11 22-172 23-157 22-3-113 21-7-13 58 2-232 2-3-193 2-17-37 2-7-97 2-3« 60 22-5-53 23-5-29 22-32.5-7 2^-5-17 . 22-5-73 62 2-32-59 2-7-83 2-631 2-3-227 2-17-43 64 23-7-19 22-3-97 2' -79 22-11-31 23-3-61 66 2-13-41 2-11-53 2-3-211 2-683 2-7*33 68 22-3-89 21-73 22-317 23-32-19 22-367 70 2-5-107 \ 2^-67 2-32-5-13 2-5-127 2-5-137 2-3-5-72 72 22-293 23-3-53 22.73 28-23 74 2-3-179 2-587 2-72-13 2-3-229 2-11-67 76 22-269 23-3-72 22-11-29 25-43 22-32-41 78 2-72-11 2-19-31 2-32-71 2-13-53 2-739 80 23-33-5 22-5-59 2«-5 22-3-5-23 23-5-37 82 2-541 2-3-197 2-641 2-691 2-3-13-19 84 22-271 25-37 22.3-107 23-173 22-7-53 86 2-3-181 2-593 2-643 2-32-7-11 2-743 88 28-17 22.33-11 23-7-23 22-347 21-3-31 90 2-5-109 2-5-7-17 2-3-5-43 2-5-139 2-5-149 92 22-3-7-13 23-149 22-17-19 2^-3-29 - 22-373 91 2-547 2-3-199 2-647 2-17-41 2-32-83 96 23-137 22-13-23 2'-3* 22-349 23-11-17 98 2-32-61 2-599 2-11-59 2-3-233 2-7-107 100 22-52-11 2^-3-52 22-52.13 23-52-7 22-3-53 OF EVEN NUMBERS. 141 From 1,500 1,600 1,700 1,800 1,900 ■ to 1,600 1,700 1,800 1,900 2,000 2 2-751 2-3»-89 2-23-37 2-17-53 2-3-317 4 2*- 47 2=-401 23-3-71 22-11-41 2<-7-17 6 2-3-251 2-11-73 2-853 2-3-7-43 2-953 8 2'-13-29 23-3-67 22-7-61 2^-113 22-32-53 10 2-5-15f 2-5-7-23 2-32-5-19 2-5-181 2-5-191 12 33.33.7 2^-13-31 2^-107 • 22-3-151 23-239 14 2-757 2-3-269 ■ 2-857 2-907' 2-3-11-29 16 2''-379 2^-101 22-3-11-13 23-297 22-479 18 2-3-11-23 2-809 2-859 2-32-101 2-7-137 20 2^-5-19 22-3''-5 23-5-43 22-5-7-13 2'-3-5' 22 2-761 2-811 2-3-7-41 ■ 2-911 2-312 24 2=-3-127 23-7-29 22-431 2^-3-19 22-13-37 26 2-7-109 2-3-271 2-863 2-11-83 2-32-107 28 23-191 2^-11-37 2«-33 22-457 23-241 , 30 2-3«-5-17 2-5-163 2-5-173 2-3-5-61 2-5-193 32 22-383 25-3-17 22-433 23-229 22-3-7-23 34 2-13-59 2-19-43 2-3-172 2-7-131 2-967 36 29-3 • 2i'-409 23-7-31 22-33-17 2^-112 38 2-769 2- 3''- 7- 13 • 2-11-79 2-919 2-3-17-19 40 2=-5-7-ll 23-5-41 22-3-5-29 2''-5-23 22-5-97 42 2-3-257 2-821 2-13-67 2-3-307 2-971 44 2^-193 2''-3-137 2*- 109 22-461 33.35 46 2-773 2-823 2-32-97 2-13-71 2-7-139 48 2=-3'-43 2^-103 22-19-23 23-3-7-11 22-487 SO 2-5'-31 2-3-5=-ll 2-53-7 2-52-37 2-3-52-13 52 2<-97 2i'-7-59 23-3-73 22-463 26-61 54 2-3-7-37 2-827 2-877 2-32-103 2-97? 56 2'-389 23-3^-23 22-439 2«-29 22-3-163 58 2-19-41 2-829 2-3-293 2-929 2-11-89 60 23-3-5-13 2'-5-83 2S-5-11 22-3-5-31 33.5-72 62 2-11-71 2-3-277 2-881 2-72-19 2-32-109 64 2''-17-23 2'- 13 22-32-72 23-233 22-491 66 2-33-29 2-7''-17 2-883 2-3-311 2-983 68 25.72 2^-3-139 23-13-17 32-467 2«-3-41 70 2-5-157 2-5-167 2-3-5-59 2-5-11-17 2-5-197 72 2'i-3-131 23-11-19 22-443 2^-32-13 22-17-29 74 2-787 2-33-31 2-887 2-937 3-3-7-47 76 23-197 2^-419 2*-3-37 22-7-67 3'-13-19 78 2-3-263 2-839 2-7-127 2-3-313 2-33-43 80 2'-5-79 2''-3-5-7 22-5-89 23-5-47 22-32-5-11 82 2-7-113 2-29« 2-3*- 11 9-941 3-991 84 2'-3=-ll 2^-421 23-293 22-3-157 2»-31 86 2-13-61 2-3-281 2-19-47 2-23-41 3-3-331 88 2'- 397 23-211 22-3-149 2^-59 32-7-71 90 2-3-5-53 2-5- 13= 2-5-179 2-33-5-7 2-5-109 • 92 23-199 32. 3^-47 28-7 22-11-43 23-3-83 94 2-797 2-7-lP 2-3-13-23 2-947 2-997 96 22-3-7-19 2^5-53 22-449 23-3-79 22-499 98 2-17-47 ■ 2-3-283 2-29-31 2-13-73 2-33-37 100 2«-52 32-52-17 33.32-52 22-52-19 94.53 ' 142 PRIiME FACTORS From 2,000 2,100 2,200 2,300 2,400 to 2,100 2,200 2,300 2,400 2,500 2 2-7-11-13 2-1051 2-3-367 2-1151 2-1201 4 2«-3-167 23-263 22-19-29 2«-3« 22-601 6 2-17-59 2-3^-13 2-1103 2-1153 2-3-401 8 23-251 2«- 17-31 2^-3-23 22-577 23.7.43 10 2>3-5-67 2-5-211 2-5-13-17 2-3-5-7-11 2.5-241 12 2=-503 2»-3-ll 2S-7-79 23-172 22- 32. 67 14 2-19-53 2-7-151 2-33-41 2-13-89 2.17.71 16 2*-3=-7 2»-23= • 23-277 22-3-193 2<'.151 18 2-1009 2-3-353 2-1109 2-19-61 2.3.13.31 20 2«-5-101 23-5-53 22-3-5-37 2''-5-29 22.5.112 22 2-3-337 2-1061 2-11-101 2-33-43 2.7.173 24 2=-ll-23 2«-3«-59 2*- 139 22-7-83 23.3.101 26 2-1013 2-1063 2-3-7-53 2-1163 2.1213 28 2»-3-13'' 2''-7-19 22-557 23-3-97 22.607 30 2-5-7-29 2-3-5-71 2.5-223 2-5-233 2.3^.5 32 a*- 127 2^-13-41 23-32-31 22-11-53 2'. 19 34 2-3i'-113 2-11-97 2-1117 2-3-389 2.1217 36 2'- 509 23-3-89 22-13-43 2" -73 22.3.7.29 38 2-1019 2-1069 2-3-373 2-7-167 2.23.53 40 2^-3-5-17 2=-5-107 2«-5-7 22-32-5-13 23.5.6I 42 2-1021 2-3'-7-17 2-19-59 2-1171 2.3.11.37 44 2=-7-73 2^-67 22.3-11-17 23-293 22.13.47 46 2-3-11-31 2-29-37 2-1123 2-3-17-23 2.1223 48 2" 2^-3-179 23-281 22-587 2^-32-17 50 2.5^-41 2-5=-43 2-32.53 2-52-47 2-52-72 52 22.33.19 23-269 22-563 34.3.72 22-613 54 2-13-79 2-3-359 2-72-23 2.11.107 2-3-409 56 23-257 2''-7=-ll 2^-3-47 22.19-31 23-307 58 2-3-73 2-13-83 2-1129 2-32-131 2-1229 60 22-5-103 2^-33-5 22-5-113 23.5.59 22.3.5.41 62 2-1031 2-23-47 2-3-13-29 2.1181 2.1231 64 2''-3-43 2»-541 23-283 22.3.197 2*.7.11 66 2-1033 2-3-19=^ 2-11-103 2.7.132 2.32.137 68 2'-ll-47 23-271 22-3^-7 2«.37 22.617 70 2-3''-5-33 2-5-7-31 2-5-227 2.3.5.79 2.5.13.19 72 2'-7-37 2=-3-181 2^-71 22.593 23.3.103 74 2-17-61 2-1087 2-3-379 2.1187 2.1237 76 2'-3-173 2'- 17 22-569 . 23.33.11 22.619 78 2-1039 2-3=-lP 2-17-67 2.29.41 2-3.7.59 80 2^-5-13 2»-5-109 23-3-5-19 22.5.7.17 2^.5.31 82 2-3-347 2-1091 2-7-163 2.3.397 2.17.73 84 2'=- 521 23-3-7-13 22-571 2*. 149 22.33.23 86 2-7-149 2-1093 2-32-127 2.1193 2*. 11. 113 88 23-3^-29 2" -547 a''. 11-13 22.3.199 23.311 90 2-5-11-19 2-3-5-73 2-5-229 2.5.239 2.3.5.83 92 2=^-523 2"- 137 22-3-191 23.13.23 22.7.89 94 2-3-349 2-1097 2-31-37 2-32-7-19 2.29.43 96 2<-131 2»-3=-61 23-7-41 22-599 2».3.13 98 2-1049 2-7-151 2-3-383 2-11-109 2-1249 100 2'-3-5'=-7 23-5'- 11 22-52-23 2^-3-52 22.5* OF EVEN NUMBERS. 143 From 2,500 2,600 2,700 2,800 2,900 to 2,600 2,700 2,800 2,900 3,000 2 2-3^-139 2-1301 2-7-193 2-3-467 2-1451 4 2»-313 2'-3-7-31 2^-13'i 22-701 23-3-112 6 2-7-179 2-1303 2-3-11-41 2-23-61 2-1453 8 2=-3-ll-19 a'' -163 22-677 23-33-13 22-727 10 2-5-251 2-3^-5-29 2-5-271 2-5-281 2-3-5-97 12 2*- 157 2=i-653 23-3-113 22-19-37 33-7-13 14 2-3-419 2-130T 2-23-59 2-3-7-67 2-31-47 16 2'-17-37 23-3-109 22-7-97 2»-ll 22 -3« 18 2-1259 2-7-11-17 2-32-151 2-1409 2-1459 20 23-3«-5>7 2^-5-13] 2'- 5- 17 22-3-5-47 23.5-73 22 2-13-97 2-3-19-23 2-1361 2-17-83 2-3-487 24 2^-631 21= -41 22-3-227 23-353 22-17-43 26 2-3-421 2-13-101 2-29-47 2-32-157 2-7-11-19 28 2^-79 22-3^-73 23-11-31 22- T- 101 2''-3-61 30 2-5-11-23 2-5-263 2-3-5-7-13 ■ 2-5-283 2-5-293 32 2=-3-211 23-7-47 22-683 2''-3-59 22-733 34 2'- 7- 181 2-3-439 2-1367 2-13-109 a-32-163 36 23-317 2'- 659 2''-32-19 22-709 23-367 38 2-33-47 2-1319 2-372 2-3-11-43 2-13-113 40 2^-5-127 2^-3-5-11 22-5-137 23-5-71 22-3-5-72 42 2-31-41 2-1321 2-3-457 2-72-29 2-1471 44 2^-3-53 2=-661 23-73 22-32-79 2' -23 46 2-19-67 2-33-72 2-1373 2-1423 2-3-491 48 2«-7''-13 23-331 22-3-229 2's-89 22-11.67 50 2-3-5^-17 2-52-53 2-53-11 2-3-52-19 2-52.59 52 2=-ll-29 22-3-13-17 2«-43 22-33-31 23.32.41 54 2-1277 2-1327 2-3"- 17 a- 1427 2-7-211 56 2''-3'-71 2^-83 22-13-53 a3-3-7-17 22-739 58 2-1279 2-3-443 2-7-197 a- 1429 2.3.17.29 60 2''-5 22-5-7-19 23-3-5-23 22-5-11-13 2". 5- 37 62 2-3-7-61 2-113 2-1381 2-33-53 2-1481 64 2' -641 23-32-37 22-691 2"- 179 22-3-13-19 66 a -1283 2-31-43 2-3-461 2-1433 2-1483 68 2^-3-107 22-23-29 ■a"- 173 22-3-239 23-7-53 70 2-5-257 2-3-5-89 2-5-277 2-5-7-41' 2-33-5-11 72 22-643 2^-167 22-32-7-11 23-359 22-743 74 2-3''-H-13 2-7-191 2-19-73 2-3-479 2-1487 76 2*- 7-23 22-3-223 23-347 22-719 2^-3-31 78 2- 1289 2-13-103 2-3-463 2-1439 2-1489 80 2«-3-5-43 2'-5-67 22-5-- 139 26.32.5 22-5-149 82 2-1291 2-32-149 2-13-107 2.11-131 2-3-7.71 84 23-17-19 22-11-61 2=^-3-29 22-7-103 23-373 86 2-3-431 2-17-79 2-7-199 2-3-13-37 2-1493 88 2'- 647 2''-3-7 22-17-41 23-192 22-32-83 90 2-5'7-37 2-5-269 2-32-5-31 2-5-172 2-5-13-23 92 25-3* 22-673 23-349 22-3-241 2"-ll-17 94 2- 1297 2-3-449 2-11-127, 2-1447 2-3-499 96 2^-11-59 23-337 22-3-233 2'' -181 ' 22-7-107 98 2-3-433 2-19-71 2-1399 2-32-7-23 2-1499 100 23-6^-13 22-33-52 2". 52-7 22-52-29 23-3-53 144 PRIME FACTORS From 3,000 3,100 8,200 3,300 3,400 to 3,100 3,200 3,300 3,400 3,500 2 2- 19-79 2-3- 11-47 2-1601 2-13-127 2-3»-7 4 22-751 2's-97 22-32-89 23-7-59 22-23-37 6 2-3'-167 2-1553 2-7-229 2-3-19-29 2-13-131 8 2«-47 22-3-7-37 23-401 22-827 21-3-71 10 2-5-7-43 2-5-311 2-3-5-107 2-5-331 2-5-11-31 12 2^-3-251 23-389 22-11-73 21-32-23 22-853 14 2-11-137 2-32-173 2-1607 2-1657 2-3-569 16 23-13-29 22-19-41 2''-3-67 22-829 23-7-61 18 2-3-503 2-1559 2-1609 2-3-7-79 2-1709 20 2>'-5-151 2'-3-5-13 22-5-7-23 23-5-83 22-32-5-19 22 2-1511 2-7-223 2-32-179 2-11-151 2-29-59 24 2*-3'-7 22-11-71 23-13-31 22-3-277 2^-107 26 2-17-89 2-3-521 2-1613 2-1663 2-3-571 28 22-757 23-17-23 22-3-269 2S-13 22-857 30 2-3-5-101 2-5-313 2-5-17-19 2-32-5-37 2-5-73 32 2' -379 22-33-29 2^-101 22-72-17 93-3-11-13 34 2-37-41 2-1567 2-3-72-11 2-1667 2-17-101 36 2^-3-11-23 28-72 22-809 23-3-139 22-859 38 2-72-31 2-3-523 2-1619 2-1669 2-32-191 40 2^-5-19 22-5-157 23-3*-5 22-5-167 21-5-43 42 2-32-132 2-1571 2-1621 2-3-557 2-1721 44 22-761 23-3-131 22-811 21-11-19 22-3-7-41 46 2-1523 2-112-13 2-3-541 2-7-239 2-1723 48 2^-3-127 2^-787 2^-7-29 22-33-31 23-431 50 2-52-6I 2-32-52-7 2^-53-13 2-52-67 2-3-52-23 52 22-7-109 2^-197 22.3-271 23-419 22-863 54 2-3-509 2-19-83 2-1627 2-3-13-43 2-11-157 56 2^-191 22-3-263 23-11-37 22-839 2''-33 58 2-11-139 2-1579 2-32- 181 2-23-73 2-7-13-19 60 22-32-5-17 23-5-79 22-5-163 2>-3-5-7 .22-5-173 62 2-1531 2-3-17-31 2-7-233 2-412 2-3-577 64 23-383 22-7-113 2«-3-17 22-292 23-433 66 2-3-7-73 2-1583 2-23-71 2-32-11-17 2-1733 68 22-13-59 2^-32-11 22-19-43 23-421 22-3-172 70 2-5-307 2-5-317 2-3-5-109 2-5-337 2-5-347 72 2"'-3 22-13-61 23-409 ' 22-3-281 21-7-31 74 2-29-53 2-3-232 2-1637 2-7-241 2-32-193 76 22-769 23-397 22-32-7-13 21-211 22-1] -79 78 2-3''-19 2-7-227 2-11-149 2-3-563 2-37-47 80 2»-5-7-ll 22-3-5-53 2^-5-41 22-5-132 23-3-5-29 82 2-23-67 2-37-43 2-3-547 2-19-89 2-1741 84 22-3-257 2"- 199 22-821 23-32-47 22-13-67 86 2-1543 2-33-59 2-31-53 2-1693 2-3-7-83 88 2^-193 22-797 23-3-137 22-7-lP 2' -109 90 2-3-5-103 2-5-11-29 2-5-7-47 2-3-5-113 2-5-349 92 22-773 23-3-7-19 22-823 2»-53 22-32-97 94 2-7-13-17 2-1597 2-33-6I 2-1697 2-1747 96 23-32-43 22-17-47 2^-103 22-3-283 23-19-23 98 2-1549 2-3-13-41 2-17-97 2-1699 2-3-11-53 100 22-52-31 2' -52 22-3-52-11 23-52-17 22-53.7 OP EVEN NUMBERS. 145 From 3,500 , 3,600 3,700 3,800 3,900 to 3,600 3,700 3,800 3,900 4,000 2 2- 17 -103 2-1801 2-3-617 2-1901 2-1951 4 2*- 3 -73 2^-17-53 23-463 22-3-317 28-61 6 2-1753 2-3-601 2-17-109 2-11-173 2-32-7-31 8 2=-877 23-11-41 22-32-103 25-7-17 22-977 10 2-33-5-13 2-5-19=i 2-5-7-53 2-3-5-127 2-5-17-23 12 23-439 2=-3-7-43 a^'-sg 22-953 23-3-163 14 2-7-251 2-13-139 2-3-619 2-1907 2-19-103 16 2=-3-293 25-113 22-929 23-32-53 22-11-89 18 2-1759 2-33-67 2-11-132 2-23-83 2-3-653 20 2»-5-ll 2'-5-181 23-3-5-31 22-5-191 2^-5-72 22 2-3-587 2-1811 2-1861 2'3-72-13 2-37-53 24 2''-881 23-3-151 22-72-19 2''-239 22-32-109 26 2- 41 -43 2-72-37 2-3«-23 2-1913 2-13-151 28 33.32.7!! 2^-907 2^-233 22-3-11-29 23-491 30 2-5-353 2-3-5-lP 2-5-373 2-5-383 2-3-5-131 32 25- 883 2" -277 22-3-311 23-479 22-983 34 2-3-19-31 2-23-79 2-1867 2-33-71 2-7-281 36 2*-13-17 2=-3''-101 23-467 22-7-137 2'-3-41 38 a-29-61 2-17-107 2-3-7-89 2-19-101 2-11-179 40 2»-3-5-59 23-5-7-13 22-5-11-17 28-3-5 22-5-197 42 2-7-11-23 2-3-607 2-1871 2-17-113 2-33-73 44 23-443 2^-911 2^-32-13 22-312 23-17-29 46 2-3^.197 2-1823 2-1873 2-3-641 2-1973 48 2'*- 887 a^-s-ig 22-937 23-13-37 22-3-7-47 50 2-5=-71 2-52-73 2-3-5'' 2-52-7-11 2-52.79 52 25-3-37 22-11-83 23-7-67 22-32-107 2^-13-19 54 2-177T 2-32-7-29 9rl877 2-41-47 2-3-659 56 22- 7- 127 23-457 22-3-313 2''-241 22-23:43 58 2-3-593 2-31-59 2-1879 2-3-643 2-1979 60 23-5-89 2=-3-5-61 2-'-5-47 22-5-193 f .23-32-5-11 62 2-13-137 2-1831 2-32-11-19 2-1931 2-7-283 64 22-3^-ll 2^-229 22-941 23-3-7-23 22-991 66 2-1783 2-3-13-47 2-7-269 2-1933 2-3-661 68 2* -223 22-7-131 ■ 23-3-157 22-967 2'-31 70 2-3-5-7-17 2-5-367 2-5-13-29 2-32-5-43 2-5-397 72 22- 19- 47 23-33-17 22-23-41 2^-112 22-3-331 74 2-1787 2-11-167 2-3-17-37 2-13-149 2-1987 76 23-3-149 22-919 28-59 22-3-17-19 23-7-71 78 2-1789 2-3-613 2- 1889 2-7-277 2-32-13-17 80 22-5-179 25-5-23 22-33-5-7 23-5-97 32.5.199 82 2'3^-199 2-7-263 2'31-61 2-3-647 2-11-181 84 2'- 7 , 22-3-307 23-11-43 22-971 2*-3-83 86 2-11-163 2-19-97 2-3-631 2-29-67 2-1993 88 22-3- 13-23 23-461 22-947 ■ 2^-35 22-997 90 2-5-359 2-32-5-41 2-5-379 2-5-389 2-3-5-7-19 92 23-449 22-13-71 2''-3-79 22-7-139 23-499 94 2-3-599 2-1847 2-7-271 2-3-11-59 2-1997. 22-33-^ 96 2«-29-31 2*-3-7-ll 22-13-73 23-487 98 2-7-257 2-432 2-32-211 2-1949 2-1999 100 2'-3=-52 22-52-37 23-52-19 22-3-52-13 2" -.53 19 146 PRIME FACTORS Prom 4,000 4,100 4,200 4,300 4,400 to 4400 4,200 4,300 4,400 4,500 2 2-3-23-29 2-7-293 2-11-191 2-3=-239 2-31-71 4 s'i-T-nas 2=-3'-19 ■ 2»-1051 2''-269 2«-3-367 6 2-2003 2-2053 2-3-701 2-2153 2-2203 8 2'-3-167 2«-13-79 2<-263 2=-3-359 23-19-29 10 2.5-401 2-3-5-137 2-5-421 2-5-431 2-3=-5-7« 12 a=-17-59 2*-257 2»'3<-13 23-7=- 11 2=-1103 14 a-s'-sas 2- 11=- 17 2-7=-43 2-3-719 2-2207 16 2^-251 2«-3-7» 2»- 17-31 2=-13-83 2«-3-23 18 2-7»-41 2-29-71 2-3-19-37 2-17-127 2-47= 20 2=-3-5-67 23'5-103 2»-5.211 25-33-5 2«-5-13-17 22 2-20H 2-3=-229 2-2111 2-2161 2-3-11-67 24 2' -503 2»-1031 2'-3-ll 2=-23-47 -23-7-79 26 2-3-11-61 2-2063 2-2113 2-3-7-103 2-2213 28 2»-19-53 2^-3-43 2«-7-151 23-541 2»-33-41 30 2-5-13-31 2-5-7-59 2-3=-5-47 2-5-433 2-5-443 32 2«-3«-7 2=-1033 2^-23= 22.3.192 2" -277 34 2-2017 2-3-13-53 2-29-73 2-11-197 2-3-739 36 2«-1009 2'- 11-47 2»-3-353 2^-271 2= -1109 38 2-3-673 2-2069 2-13-163 2-3=-241 2-7-317 40 2'-5-101 2'-3=-5-23 2*- 5- 53 2=-5-7-31 2»-3-5-37 42 2-43-47 2-19-109 2-3-7-101 2-13-167 2-2221 44 2=-3-337 2^-7-37 2= -1061 23-3-181 2=-ll-10] 46 2-7-17= 2-3-691 2-11-193 2-41-53 2-3=-13-19 48 2*-ll-23 2=-17-61 23-3=-59 2= -1087 2^-139 50 2-3^-52 2-5»-83 2-53-17 2-3-5=-29 2-5=-89 ,52 2= -1013 2'-3-173 2=-1063 28-17 2=-3-7-53 54 2-2027 2-31-67 2-3-709 2-7-311 '2-17-131 56 23-3-13' 2= -1039 2'>-7-19 22.32.112 23-557 58 2-2029 2-33-7-11 2-2129 2-2179 2-3-743 60 22-5-7-29 2'=-5-13 2«-3-5-71 23-5-109 2»-5-223 62 2-3-677 2-2081 2-2131 2-3-727 2-23-97 64 2^-127 2=-3-347 23-13-41 2«-1091 2*-3»-31 66 2-19-107 2-2083 2-33-79 2-37-59 2-7-11-29 68 2'-3'-113 2' -521 2»-ll-97 . 2*-3-7-13 2»-1117 70 2-5-11-37 2-3-5-139 2-5-7-61 2-5-19-23 2-3-5-149 72 2^-509 2«-7-149 2^.3-89 2»-1093 23-13-43 74 2-3-7-97 2-2087 2-2137 2-3' 2-2237 76 2'- 1019 2<-3=-29 2=-1069 23-547 2'-3-373 78 2-2039 2-2089 2-3-23-31 2-11-199 2-2239 80 2^-3-5-17 2=-5-ll-19 23-5-107 2»-3-5-73 2'-5-7 82 2-13-157 2-3-17-41 2-2141 2-7-313 2-33-83 84 2«-1021 2' -523 2«-3»-7-17 2' -137 2«-19-59 86 2-3»-227 2-7-13-23 2-2143 2-3-17-43 2-2243 88 2'- 7- 73 2=-3-349 2«-67 2"- 1097 23-3-11-17 90 2-5-409 2-5-419 2-3-5-11-13 2-5-439 2-5-449 92 2'-3-ll-31 2^-131 2»-29-37 23-3=-61 2»-]123 94 2-23-89 2-3=-233 2-19-113 2-133 2-3-7-107 96 21! 2=-1049 23-3-179 2»-7-151 2*-281 98 2-3-683 2-2099 2-7-307 2-3-733 2-13-173 100 2'-5'-41 2'-3-5«.7 2«-5«-43 2*-5=-ll 2=-3=-53 OB-" EVEN NUMBERS. 147 From 4,500 4,600 4,700 4,800 4,900 to 4,600 4,700 4,800 4,900 5,000 2 2-2251 2-3-13-59 2-2351 2-7* 2-3-19-43 4 2' -sea 22-1151 2S-3-72 22-1201 23-613 6 2-3-751 2-72-47 2-13-181 2-33-89 2-11-223 8 2«-7'-23 2»-32 22-11-107 23-601 22-3-409 10 2-5-H-41 2-5-461 2-3-5-157 2-5-13-37 2-5-491 12 2S.3-47 22-1153 23-19-31 22-3-401 2<-307 14 2-37-61 2-3-769. 2-2357 2-29-83 2-33-7-13 16 2"- 1199 23-577 22-32-131 2*-7-43 22-1229 18 2-3«-251 2-2309 2-7-337 2-3-11-73 2-2459 20 2'-5-113 22-3-5-7-11 2^-5-59 22-5-241 23.3.5.41 22 2-7-17-19 2-2311 2-3-787 2-2411 2.23-107 24 2'-3-13-29 2*- 172 22-1181 23-32-67 22. 1231 26 2-31-73 2-32-257 2-17-139 2-19-127 2-3.821 28 2*-283 22-13-89 23-3-197 22-17-71 2».7.11 30 2-3'5'151 2-5-463 2-5-11-43 2-3-5-7-23 2.5.17-29 32 2=-lM03 23-3-193 22-7-132 2= -151 22.32.137 34 2-2267 2-7-331 2-32-263 2-2417 2-2467 36 2'-3<-7 22-19-61 2'-37 22-3-13-31 23-617 38 2-2269 2-3-773 2-23-103 2-41-59 2-3-823 40 2«-5-227 2^-5-29 22-3-5-79 23-5-112 22-5-13-19 42 2-3-757 2-11-211 2-2371 2-32-269 2-7-353 44 2«-71 22-33-43 23-593 22-7-173 2^-3-103 46 2-2273 2-23-101 2-3-7-113 2-2423 2-2473 48 22-3-379 23-7-83 22-1187 2*-3-101 22-1237 50 2-5=-7-13 •2-3-52-31 2-53-19 2-52-97 2-32-52-11 52 23-569 22-1163 2*. 33-11 22-1213 23-619 54 2-32-11-23 2-13-179 2-2377 2-3-809 2-2477 56 22-17-67 2^-3-97 22-29-41 23-607 22.3-7-59 58 2-43-53 2-17-137 2-3-13-61 2-7-347 2-37-67 60 2*-3-5-19 22-5-233 23-5-7-17 22-3^-5 25-5-31 62 2-2281 2-32-7-37 2-2381 2-11-13-17 2-3-827 64 22-7-163 23-11-53 22-3-397 28-19 22-17-73 66 2-3-761 2-2333 2-2383 2-3-811 2-13-191 68 2^-571 22-3-389 2= -149 22-1217 23-33-23 70 2-5-457 2-5-467 2-32-5-53 2-5-487 2-5-7-71 72 2'-3=-127 2=-73 22-1193 23-3-7-29 22-11.113 74 2-2287 2-3-19-41 2-7-11-31 2-2437 2-3-829 76 2^-11-13 22-7-167 23-3-199 22-23-53 2*-311 78 2-3-7-109 2-2339 2-2389 2-32-271 2-19-131 80 22-5-229 23-32-5-13 22-5-239 2<-5-61 22-3-5-83 82 2-29-79 2-2341 2-3-797 2-2441 2-47-53 84 23-3-191 22-1171 2<-13-23 22-3-11-37 23-7-89 86 2-2293 2-3-11-71 2-2393 2-7-349 2-- 32 -277 88 22-31-37 2«-293 22-32-7-19 23-13-47 22-29-43 90 2.33-5-17 2-5-7-67 2-5-479 2-3-5-163 2-5-499 92 2*-7-41 22-3-17-23 23-599 22-1223 2'-3-13 94 2-2297 2-2347 2-3-17-47 2-2447 2-11-227 96 22-3-383 23-587 22-11-109 2^-32-17 22-1249 98 2- 112-19 2-3^-29 2-2399 2-31-79 2-3-72-17 100 23-52-23 22-52-47 2«-3-52 22.52-72 23-5" 148 PRIME FACTORS Proin 5,000 5,100 5,200 5,300 5,400 to 5,100 5,200 5,300 5,400 5,500 2 r 2-41-61 2-2551 2-3=- 17' 2-11-241 2-37-73 4 2'=-3''-139 2*- 11-29 2'- 1301 23-3-13-17 2'- 7- 193 6 2-2503 2-3-23-37 2-19-137 2-7-379 2-3-17-53 8 2^-313 2'- 1277 23-3-7-31 2' -1327 25-13' 10 2-3-5-167 2-5-7-73 2-5-521 2- 3'- 5- 59 2-5-541 12 2'-7-179 23-3'-71 2' -1303 2«-83 2'.3-ll-41 14 2-23-109 2-2557 2-3-11-79 2-2657 2-2707 16 2»-3-ll-19 2' -1279 2^-163 2'-3-443 23-677 18 2- 13- 193 2-3-853 2-2609 2-2659 2-3'.7.43 20 22-5-251 210-5 2'-3'-5-29 23-5-7-19 2«-5.271 22 2-3^-31 2-13-197 2-7-373 2-3-887 2-2711 24 25-157 2'-3-7-61 33-653 2'- 113 2^-3-113 26 2-7-359 2-11-233 2-3- 13-67 2-2663 2-2713 28 2^-3-419 23-641 2'- 1307 2*-3'-37 2'- 23 -59 30 2-5-503 2-33-5-19 2-5-523 2-5-13-41 2-3-5-181 32 2=-17-37 2'- 1283 2^-3-109 2'-31-43 23-7-97 34 2-3-839 2-17-151 2-2617 2-3-7-127 2-11-13-19. 36 22-1259 2^-3-107 2'-7-ll-17 23-23.29 2'-3'-151 38 2-11-229 2-7-367 2-33-97 2-17-157 2-2719 40 2<-3^-5-7 2'.5-257 23-5-131 2'-3-5-89 28-5-17 42 3-2521 2-3-857 2-2621 2-2671 2-3-907 44 2=-13-97 23-643 2'-3- 19-23 25-167 2'- 1361 46 2-3-29' 2-31-83 2-43-61 2-35-11 >• 2-7-389 48 2= -631 2'-3'-ll-13 2'-41 2'- 7- 191 23-3-227 SO 2-5'-101 2- 5' -103 2-3-53-7 2-5'-107. 2-5'- 109 52 2«-3-421 25-7-23 2' -13 -101 23-3-223 2'-29-47 54 2-7-19' 2-3-859 2-37-71 2-2677 2-33-101 56 2«-79 2'- 1289 23-3'-73 2'-13-103 ' 2«.11-31 58 2-3»-281 2-2579 2-11-239 2-3-19-47 2-2729 60 2'-5-ll-23 23-3-5-43 2'-5-263 2^-5-67 2'-3-5-7-13 62 2-2531 2-29-89 2-3-877 2-7-383 2-2731 64 23-3-211 2'- 1291 2^-7-47 2'-3'-149 23-683 66 2-17-149 2-3'-7-41 2-2633 2-2683 2-3-911 68 2'- 7 -181 2^-17-19 2'-3-439 2-''-ll-61 2'- 1367 70 2-3-5-13' 2-5-11-47 2-5-17-31 2-3-5-179 2-5-547 72 2*- 317 2'-3-431 23-659 2'-17-79 25-3'- 19 74 2-43-59 2-13-199 2-3'-293 2-2687 2-7-17-23 76 22-33.47 23-647 2'- 1319 28-3-7 2'-37' 78 2-2539 2-3-863 2-7-13-29 2-2689 2-3-11-83 80 2'-5-127 2'-5-7-37 25.3-5-11 2'-5-269 23-5-137 82 2-3-7-11' B-2591 2-19-139 2-3'- 13-23 2.2741 84 2'-31-41 2«-3* 2'. 1321 23-673 2'.3.457 86 2-2543 2-2593 2-3-881 2.2693 2-13-211 88 2»-3-53 2' -1297 23-661 2'.3-449 34.73 90 2-5-509 2-3-5-173 2-5-23' 2-5-7'-ll 2.3'-5-61 92 2'- 19 -67 23-11-59 22.33.7s 2*-337 2'- 1373 94 2-3'-283 2-7'-53 2-2647 2-3-29-31 2-41-67 96 23 -7'- 13 22-3-433 2«-331 2'.19.71 23-3-229 98 2-2549 2-23-113 2-3-883 ■ 2-2699 2-2749 100 2'-3-5'-17 2^-5'-13 2'-5'-53 23-33-5' 2'-53-ll OP EVEN NUMBERS. 149 From 5,500 5,600 5,700 5,800 5,900 to 5,600 5,700 5,800 5,900 6,000 2 2-3-7-131 2-2801 2-2851 2-3-967 2-13-227 4 2'- 43 2'-3-467 23-23-31 22-1451 2''-32-41 6 2-S7-53 2-2803 ' 2-32-317 2-2903 2-2953 8 22-3''-17 2^-701 22-1427 2<-3-112 32-7-211 10 2-5- 19-29 2-3-5-11-17 2-5-571 2-5-7-83 2-3-5-197 12 2'- 13-53 2''-23-6] 2''-3-7-17 22-1453 2= -739 ,i 14- 2-3-919 2-7-401 2-2857 2-32-17-19 2-2957 16 2'-7-197 2«-3'-13 22-1429 23-727 22-3-17.29 18 2-31-89 2-532 2-3-953 2-2909 2'1].269 20 2*-3-5-23 22-5-281 23-5-11-13 22-3-5-97 25-5-37 22 2-11-251 2-3-937 9-2861 2-41.71 2-32-7-47 24 2»-1381 23-19-37 22-33-5a 28-7-13 22-1481 26 2-3'-307 2-29-97 2-7-409 2-3-971 2-2963 28 2^-691 2''-3-7-67 2^-179 22-31-47 23-3-13-19 30 2-5-7-79 2-5-563 2-3^5-191 2-5-11-53 2-5-593 , 32 2^^3-461 29-11 22-1433 23- 3« 22-1483 34 2-2767 2-32-313 2-47-61 2-2917 2-3-23-43 36 25-173 22-1409 23-3-239 22-1459 2''-7-53 38 2-3-13-71 2-2819 2-19-151 2-3-7-139 2-2969 40 2»-5-277 23-3-5-47 22-5-7-41 2<-5-73 22-33-5-11 42 2-17-163 2-7-13-31 2-32-11-29 2-23-127 2-2971 44 23-3^-7-11 22-17-83 2*-359 22-3-487 23-743 46 2-47-59 2-3-941 2-132.17 2-37-79 2-3-991 48 2='-19-73 2«-353 22.3-479 23-17-43 22-1487 50 2-3-5^-37 2-52-113 2-53-23 2-32-52.1^ 2-52-7-17 52 2*- 347 22-32-157 23-719 22-7-11-19 2«-3-31 54 2-2777 2-11-257 2-3-7-137 2-2927 2-13-229 56 22-3-463 23-7-101 22-1439 2^-3-61 22-1489 58 2-7-397 '2-3-23-41 2-2879 2-29-101 2-32-331 60 2i'-5-139 22-5-283 2'-32-5 22-5-293 23-5-149 62 2-3'-103 2-19-149 2-43-67 2-3-977 2-11-271 64 2'- 13- 107 2's-3-59 22-11-131 23-733 22-3-7-71 66 2- 11^-23 2-2833 2-3-312 2-7-419 2-19-157 68 2«-3-29 22-13-109 23-7-103 22-32-163 2*-373 70 2-5-557 2-3^-5-7 2-5-577 2-5-587 2-3-5-199 72 2^-7-199 23-709 22-3-13-37 2^-367 22-1493 74 2-3-929 2-2837 2-2887 2-3-11.89 2-29-103 76 2'-17-41 22-3-11-43 2^-192 22-13-113 23-32-83 78 2-2789 2-17-167 2-33-107 2-2939 2-72-6I 80 22-3''-5-31 2*-5-71 22-5-17^ 23-3.5-72 22-5-13-23 82 2-2791 2-3-947 2.72.59 2-17-173 2-3-997 84 2^-349 22-72-29 23.3-241 22-1471 2^-11-17 86 2-3-7^-19 2-2843 2-11-263 2-33-109 2-41-73 88 2^-11-127 23-32-79 22-1447 28-23 22-3-499 90 2-5-13-43 2-5-569 2-3-5-193 2-5-19-31 2-5-599 92 2'-3-233 22-1423 2'5-181 22-3-491 23-7-107 94 2-2797 2-3-13-73 2-2897 2-7-421 2-3^-37 96 2»-1399 2S-89 22-32-7-23 23-11-67 22-1499 98 2-3''-311 2-7-11-37 2-13-223 2-3-983 2-2999 100 2S-5«-7 22-3-52-19 23-52-29 a2-52-59 2^-3-5' 150 PRIME FACTORS Prom 6,000 6,100 6,200 6,300 6,400 to 6,100 6,200 6,300 6,400 6,500 2 2-3001 2-33-113 2-7-443 2-23-137 2-3-11-97 4 2=-19-79 2»-7-109 22-3-11-47 2*- 197 22-1601 6 2-3-7-H-13 2-43-71 2-29-107 2-3-1051 2-3203 8 2'- 751 2»-3-509 2«-97 22-19-83 2'-32-89 10 2-5-601 2-5-13-47 2-3='-5-23 2-5-631 2-5-641 12 2^-3=-167 2».191 22-1553 2'-3-263 22-7-229 14 2-31-97 2-3-1019 2-13-239 2-7-11-41 2-3-1069 16 2'- 47 22-11-139 2'-3-7-37 22-1579 2^-401 18 2-3-17-59 2-7-19-23 2-3109 2-3''- 13 2-3209 20 2»-5-7-43 2^-32-5-17 22-5-311 2^-5-79 22-3-5.107 22 2-3011 2-3061 2-3-17-61 2-29-109 2-132-19 24 2^-3-251 22-1531 2«-389 22-3-17-31 2'- 11-73 26 2-23-131 2-3-1021 2-11-283 2-3163 2-3-7-17 28 2^-11-137 2^-383 22-32-173 2'- 7- 113 22-1607 30 2-3'- 5- 67 2-5-613 2-5-7-89 2-3-5-211 2-5-643 32 2*- 13-29 S2-3-7'73 2'-19-41 22-1583 2''-3-67 34 2-7-431 2-3067 2-3-1039 2-3167 2-3217 36 2«-3-503 23-13-59 22-1559 26-32-11 22-1609 38 2-3019 2-32-11-31 2-3119 2-3169 2-3-29-37 40 2'-5-151 22-5-307 25-3-5-13 22-5-317 2'-5-7-23 42 2-3-19-53 2-37-83 2-3121 2-3-7-151 2-3221 44 2^-1511 2"-3 22-7-223 2'-13-61 22-32-179 46 2-3023 2-7-439 2-32-347 2-19-167 2-11-293 48 35.33.7 22-29-53 2'-ll-71 22-3-232 2^-13-31 50 2-5»-ll« 2-3-52-41 2-5* 2-52-127 2-3-52-43 52 2^-17-89 2'- 769 22-3-521 2" -397 22-1613 54 2-3-1009 2-17-181 2-53-59 2-32-353 2-7-461 56 2^-757 22-3^-19 2^-17-23 22-7-227 2'-3-269 58 2-13-233 2-3079 2-3-7-149 2-11-17= 2-3229 60 2=-3-5-101 2''-5-7-ll 22-5-313 2'-3-5-,53 22-5-17-19 62 2-7-433 2-3-13-79 2-31-101 2-3181 2-32-359 64 2^-379 22-23-67 23-3'-29 22-37-43 26-101 66 2-3^-337 2-3083 2-13-241 2-3-1061 2-53-61 68 2=-37-41 2'-3-257 22-1567 .2^-199 22-3-72-11 70 2-5-607 2-5-617 2-3-5-11-19 2-5-72-13 2-5-647 72 2»-3-ll-23 22-1543 37.72 22-3-'-59 2^-809 74 2-3037 2.32.73 2-3137 2-3187 2-3-13-83 76 2'-'-7»-31 2^-193 22-3-523 2^-797 22-1619 78 2-3-1013 2-3089 2-43-73 2-3-1063 2-41-79 80 2»-5-19 22-3-5-103 2'-5-157 22-5-11-29 2^-3^-5 82 2-3041 2-11-281 • 2-32-349 2-3191 2-7-463 84 22.32- 13« 2»-773 22-1571 2*-3-7-19 22-1621 86 2-17-179 2-3-1031 2-7-449 2-31-103 2-3-23-47 88 2'- 761 22-7-13-17 2''-3-131 2«-1597 23-811 90 2-3-5-7-29 2-5-619 2-5-17-37 2-32-5-71 2-5-11-59 92 22-1523 2«-32-43 22-112-13 2'-17-47 22-3.541 94 2-11-277 2-19-163 2-3-1049 2-23-139 2-17-191 96 2<-3-127 22-1549 2' -787 22-3-13-41 2^-7-29 98 2-3049 2-3-1033 2-47-67 2-7-457 2.3«-192 100 2'- 5'- 61 2'-52-31 22-32-52-7 28.52 22-53-13 OP EVEN NUMBERS. 151 From 6,500 6,600 6,700 6,800 6,900 7,000 to 6,600 6,700 6,800 6,900 2 4 2-3251 2»-3-271 2-3301 22-13-127 2-3-1117 2"- 419 2-19-179 22-36.7 2-7-17-29 23-863 6 2-3253 2-32-367 2-7-479 2.41.83 2-3-1151 8 2'i-1627 2*- 7- 59 22-3-13-43 23.23.37 22-11-157 10 2-3-5-7-31 2-5-661 2-5-11-61 2-3-5-227 2-5-691 12 2^-11-37 22-3-19-29 23-839 22-13-131 28-33 14 2-3257 9-3307 2-32-373 2-3407 2-3457 16 2=-3'-181 23-827 22-23-73 25-3-71 22-7-13-19 18 2-3259 2-3-1103 2.3359 2-7-487 2-3-1153 20 23-5-163 2^-5-331 2«-3-5-7 22.5.11.31 23-5-173 22 2-3-1087 2-7-11-43 2-3361 2.32-379 2-3461 24 2^-7-233 2^^-32-23 22-412 23-853 22-3-577 26 2-13-251 2-3313 2-3-19-59 2-3413 2-3463 28 2'-3-17 22-1657 23-292 22-3-569 2''- 433 30 2-5-653 2-3-5-13-17 2-5-673 2-5-683 2-32-5-7-11 32 22-23-71 23-829 22-32-11-17 2«-7-61 22-1733 34 2-33-11= 2-31-107 2-7-13-37 2-3-17.67 2-3467 36 23-19-43 22-3-7-79 2*- 421 22-1709 23.3-172 38 2- 7- 467 2-3319 2-3-1123 2-13-263 2-3469 40 22-3-5-109 2*-5-83 22-5-337 23-32-5-19 22-5-347 42 2-3271 2-3''-41 2-3371 2-11-311 2-3-13-89 44 2<-409 22-11-151 23-3-281 22-29-59 2*-7-31 ' 46 2-3-1091 2-3323 2-3373 2-3-7-163 2-23-151 48 22-1637 23-3-277 22-7-241 28-107 2^-32-193 50 2-52-131 2-52-7-19 2-33-53 2-52-137 2-52-139 52 23-32-7-13 22-1663 2^-211 22-3-571 23-11-79 54 2-29-113 2-3-1109 2-11-307 2-23-149 2-3-19-61 56 22-11-149 2»-13 22-3-563 23-857 22-37-47 58 2-3-1093 2-3329 2-31-109 2-33-127 2-72-71 60 2^-5-41 22-32-5-37 33-5-132 22-5-73 2<-3-5.29 62 2-17-193 2-3331 2-3-72-23 2-47-73 2.592 64 22-3-547 23-72-17 22-19-89 2''-3-ll-13 22-1741 66 2-72-67 2-3-11-101 2-17-199 2-3433 2-3^-43 68 23-821 22.1667 2^-32-47 22-17-101 23-13-67 70 2-32-5-73 2-5-23-29 2-5-677 2-3-5-229 2--5-17.41 72 22-31-53 2*-3-139 22-1693 23-859 22.3-7-83 74 2-19-173 2-47-71 2-3-1129 2.7-491 2-11-317 76 2^-3-137 22-1669 23-7-112 22-32.191 2«-109 78 2-11-13-23 2-32-7-53 2-3389 2. 19. 181 2-3-1163 80 22-5-7-47 23-5-167 22-3-5-113 2^.5.43 22-5-349 82 2-3-1097 2-13:257 2-3391 2.3.31-37 2-3491 84 23-823 22-3-557 2'-53 22-1721 23-32-97 86 2-37-89 2,-3343 2-32-13-29 2-11-313 2-7.499 88 22-33-61 2^-11-19 22-1697 23-3-7-41 22.1747 90 2-5-659 2-3-5-223 2-5-7-97 2-5-13-53 2.3-5-233 92 2«-103 22-7-239 23-3-283 22-1723 2^-19-23 94 2-3-7-157 2-3347 2-43-79 2-32-383 . 2-13-269 96 2»-17-97 23-33-31 22-1699 2''-431 22-3-11-53 98 2-3299 2-17-197 2-3-11-103 2-3449 3-3499 100 23.3-52-11 22-52-67 2''-52-17 22.3.52.23 23-53-7 152 PRIME FACTORS Prom r.OOO 7,100 7,200 7,300 7,400 to 7,100 7,200 7,300 7,400 7,500 2 2- 3'- 389 2-53-67 2-13-277 2.3.1217 2.3701 4 2^- 17- 103 26-3-37 22.1801 23-11.83 22.3.617 6 2-31-113 2-11-17-19 2.3.1201 2-13-281 2.7.232 8 2^-3-73. 22-1777 2'. 17- 53 22-32-7-29 2*. 463 10 2.5-701 2-3=-5-79 2.5.7.103 2-5-17-43 2.3.5.13-19 12 2»-1753 2'-7-127 22.3. 601 2*. 457 22-17.109 14 2-3-7-167 2-3557 2.3607 2.3.23-53 2-11-337 16 2' -877 22-3-593 2^-11-41 22.31.59 23-32-103 18 2- 112-29 2-3559 2-32-401 2.3659 2-3709 20 2'-33-5-13 2<-5-89 22.5.192 23.3. 5.6I 22.5.7.53 22 2-3511 2-3-1187 2.23-157 2-7-523 2.3.1237 24 2*- 439 22-13-137 2»-3-7.43 22-1831 26.29 26 2-3-1171 2-7-509 2.3613 2.32.11.37 2-47-79 28 2''-7-251 23-3«-ll 22.13.139 25-229 22-3-619 30 2-5-19-37 2-5-23-31 2.3.5.241 2.5.733 2-5-743 32 2'-3-893 22-1783 26.113 22.3.13.47 23-929 34 2-3517 2-3-29-41 2.3617 2.19.193 2-32-7-59 36 2= -1759 25-223 22.3^.67 23.7.131 22-11-132 38 2- 3^^-17- 23 2-43-83 2.7.11.47 2-3.1223 2-3719 40 2'- 5- 11 22-3-5-7-17 2^.5-181 22.5.367 2,*-3-5-31 42 2-7-503 2-3571 2-3-17-71 2.3671 2-612 44 22-3-587 2^-19-47 22-1811 2^.33.17 22-1861 46 2-13-271 2-32-397 2-3623 2.3673 2.3.17-73 48 2'-881 22-1787 2«.3-151 22.11.167 23-72-19 50 2-3-52.47 2-52-11-13 2-53.29 2.3.52.72 2-52.149 52 22-41-43 2'- 3- 149 22.72.37 23.919 22.3<-23 54 2-3527 2-72-73 2.32.13.31 2-3677 2-3727 56 2^-32-72 22-1789 2'. 907 22-3-613 25-233 58 2-3529 2-3-1193 2.19-191 2-13-283 2-3-11-113 60 22-5-353 2'-3-179 22-3-5-112 26-5-23 22-5-373 62 2-3-11-107 2-3581 2-3631 2-32-409 2-7-13-41 64 2'- 883 22-32-199 25.277 22-7-263 23-3-311 66 2-3533 - 2-3583 2.3.7.173 2-29-127 2-3733 68 22-3-19-31 210.7 22.23-79 23-3-307 22-1867 70 2-5-7-101 2-3-5-239 2-5-727 2-5-11-67 2-32-5-83 72 25-13-17 22-11-163 2'-32-101 22-19-97 2''-46T 74 2-3'-131 2-17-211 2-3637 2-3- 1229 2-37-101 76 22-29-61 2^-3-13-23 22-17.107 2<-461 22-3-7-89 78 2-3539 2-37-97 2.3.1213 2-7-17-31 2-3739 80 2'-3-S-59 22-5-359 2^.5. 7-13 22.32.5-41 23-5-11.17 82 2-3541 2-33-7-19 2-11.331 2-3691 2-3-29-43 84 22-7-11-23 2^-449 22.3. 607 23-13-71 22-1871 86 2-3-1181 2-3593 2-3643 2-3-1231 2-19-197 88 2^-443 22-3-599 23-911 22-1847 26.32.13 90 2-5-709 2-5-719 2-3«.5 2.5.739 2.5-7.107 92 22-32-197 2'-29-31 22.1823 25.3-7-11 22-1873 94 2-3547 2-3-11-109 2.7.521 2-3697 2.3.1249 96 2' -887 22-7-257 2''.3.19 22.432 23.937 98 2-3-7-132 2-59-61 2.41.89 2.33-137 2.23.163 100 22-52-71 25.32-52 22.52.73 23.52.37 22.3.5* OP EVEN NUMBERS. 153 From 7,500 7,600 7,700 7,800 7,900 to 7,600 7,700 7,800 7,900 8,000 2 2-1P-31 2-3.7.181 2-3851 2-47.83 2-32-439 4 ^■1-61 22.1901 23-32-107 22.1951 2=-13.19 6 2-33-139 2.3803 2-3853 2.3.1301 2.59-67 . 8 2''-1877 23-3.317 22.41.47 2' -61 22.3-659 10 2-5-751 2-5.761 2.3.5.257 2-5-11-71 2.5-7.113 12 2^-3-3i3 22-11.173 2».241 22-32-7.31 23.23.43 14 .2- 13 -IT" 2-3''-47 2.7.19.29 2-3907 2.3-1319 16 2''-1879 2«-7-17 22.3-643 23-977 22-1979 18 2-3-7-179 2-13.293 2.17-227 2-3-1303 2-37-107 20 2*-5-47 22.3-5-127 23-5-193 22-5-17-23 2^-32-5.11 22 2-3761 2-37-103 2-33-11-13 2-3911 2.17.233 24 2'-3»- 11-19 23-953 22-1931 2^-3-163 22.7.283 26 2-53-71 2-3-31-41 2-3863 2-7-13-43 2.3.1321 28 23-941 22-1907 2''-3-7.23 22.19.103 23.991 30 2-3-5-251 2.5.7.109 2.5.773 2-33-5.29 2.5.13-61 32 2^-7-269 2''.32.53 22.1933 23-11-89 22-3-661 34 2-3767 2.11-347 2.3-1289 2-3917 2-3967 36 2^-3-157 22-23-83 23-967 22-3-653 2S-31 38 2-3769 2-3-19-67 2-53-73 2-3919 2-3<'-72, 40 2=i-5-13-29 23-5-191 22-32-5.43 §6.5.72 22.5-397 42 2-32-419 2-3821 2.72.79 2-3-1307 2-11-192 44 2^-23-41 22.3.72.13 28-112 22-37-53 23-3-331 46 2-7»-ll 2.3823 2-3-1291 2-3923 2-29-137 48 2i'-3-17-37 2= -239 22-13-149 23.32.109 22 -.1987 50 2-5=-151 2.32.52.17 2-53-31 2.52.157 2-3-52.53 52 2'- 59 22.1913 23-3-17-19 22-13. 151 2*.7.71 54 2-3-1259 2.43.89 2-3877 2.3.7.11-17 2.41.97 56 2^-1889 23.3.11.29 22.7.277 2*. 491 22-32-13-17 58 2-3779 2.7.547 2-32.431 2.3929 2-23-173 60 23-33.5-7 22.5-383 2*-5.97 22.S-5-131 23-5-199 62 2-19-199 2-3.1277 2-3881 2-3931 2-3-1327 64 22-31-61 2^-479 22-3-647 23.983 22-11-181 66 2-3-13-97 2.3833 2-11-353 2.32.19-23 2 -"7 -569 68 2*-ll-43 22.33.71 23-971 22-7-281 25-3-83 70 ■2-5.757 2-5-13.59 2-3-5-7-37 2-5-787 2-5.797 72 22-3-631 23-7.137 22-29-67 2=-3-41 22.1993 74 2-7-541 2.3.1279 2-132-23 2-31-127 2-32-443 76 23-947 22.19.101 2^-3* 22-11-179 23-997 78 2-3'-421 2.11.349 2-3889 2-3-13-101 2-3989 80 22-5-379 2».3.5 22-5-389 23-5-197 22.3.5-7-19 82 2-17-223 2.23.167 2.3-1297 2-7-563 2.13.307 84 2*-3-79 22.17-113 23-7-139 22-33-73 2*. 499 86 2-3793 2-'32-7.61 2-17-229 2-3943 2.3.113 88 22-7-271 23.312 22-3.11»59 2^-17.29 22-1997 90 2-3-5-11-23 2.5-769 2-5-19-41 2.3.5-263 2.5-17-47 92 23-13-73 22-3.641 2^-487 22-1973 23-33-37 94 2-3797 2-3847 2-32.433 2-3947 2-7-571 96 92.32-911 2^-13.37 22.1949 23.3.7.47 22-1999 98 2.a9'131 2-3-1283 2.7.557 2.11-359 2-3-31-43 100 a" -5^ -19 22-52-7-11 23.3-52.13 22.- 52 -79 36-5' 20 154 PRIME FACTORS From 8,000 8,100 8,200 8,300 8,400 to 8,100 8,200 8,300 . 8,400 8,500 2 2- 4001 2-4051 2-3-1367 2-7-593 2-4201 4 2«-3-23-29 2^-1013 22-7-293 a*-3-173 22-11-191 6 2-4003 2-3-7-193 2-11-373 2-4153 2-32.-467 8 23-7-11-13 2' -2027 2''-33-19 22-31-67 23-1051 10 2-3»-5-89 2-5-811 2-5-821 2-3-5-277 2-5-292 12 2= -2003 2*-3-132 22-2053 23-1039 22-3-701 14 2-4007 2-4057 2-3-372 2-4157 2-7-601 16 2''-3-167 22-2029 23-13-79 22-33-7-11 25-263 18 2-19-2U 2-3-'-ll-41 2-7-587 2-4159 2-3-23-61 20 22-5-401 23-5-7-29 22-3-5-137 2''-5-13 22-5-421 22 2-3-7-191 2-31-131 2-4111 2-3-19-73 2-4211 24 2^-17-59 22-3-677 2* -257 22-2081 23-3"- 13 26 2-4013 2-17-239 2-32-457 2-23-181 2-11-383 28 2=-3'-223 2"- 127 22-112-17 23-3-347 22-72-43 30 2-5-11-73 2-3-5-271 2-5-823 2-5-7^-17 2-3-5-281 32 2^-251 2«-19-107 23.3-73 22-2083 2''-17-31 34 2-3-13-103 2-72-83 2-23-179 2-32-463 2-4217 36 22. 7a. 41 23-32-113 22-29-71 2*-521 22-3-19-37 38 2-4019 2-13-313 2-3-1373 2-11-379 2-4219 40 2^-3-5-67 22-5-11-S7 2^-5-103 22-3-5-139 23-5-211 ^2 2-4021 2-3-23-59 2-13-317 2-43-97 2-32.7-67 44 2«-2011 2^-509 22-32-229 23-7-149 22-2111 46 2-3^-149 2-4073 2-7-19-31 2-3-13-107 2-41-103 48 2^-503 22-3-7-97 23-1031 22-2087 28-3-11 50 2-52-7-23 2-52-163 2-3-53-11 2-52-167 2-52-132 52 2^-3-11-61 23-1019 22-2063 25-32-29 22-2113 54 2-4027 2-33-151 2-4127 2-4177 2-3-1409 56 2^-19-53 22-2039 2«-3-43 22-2089 23-7-151 58 2-3-17-79 2-4079 2-4129 2-3-7-199 2-4229 60 2«-5-13-31 2'-3-5-17 22-5-7-59 23-5-11-19 22-32-5-47 62 2-29-139 2-7-11-53 2-3^-17 2-37-113 2-4231 64 2'-3''-7 22-13-157 23-1033 22-3- 17-41 2''- 232 66 2-37-109 2-3-1361 2-4133 2-47-89 2-3-17-83 68 2^-2017 23-1021 22-3-13-53 2*-523 22-29-73 . 70 2-3-5-269 2-5-19-43 2-5-827 2-33-5-31 2-5-7-112 72 23-1009 22-33-227 2'-n-47 22-7-13-23 23-3-353 74 2-11-367 2-61-67 2-3-7-197 2-53-79 2-19-223 76 2''-3-673 2*- 7 -73 22-2069 23-3-349 22-13-163 78 2-7-577 2-3-29-47 2-4139 2-59-71 2-33-157 80 2*- 5- 101 22-5-409. 23-32-5-23 22-5-419 25-5-53 82 2-3''-449 2-4091 2-41-101 2-3-11-127 2-4241 84 2^-43-47 23-3-11-31 22-19-109 2«-131 . 22-3-7-101 86 2-13-311 2-4093 2-3-1381 2-7-599 2-4243 88 2^-3-337 22-23-89 2='-7-37 22-32-233 23-1061 90 2-5-809 2-3«'5-7-13 2-5-829 2-5-839 2-3-5-283 92 22.7-172 213 22-3-691 23-1049 22-11-193 94 2-3-19-71 2-17-241 2-11-13-29 2-3-1399 2-31-137 96 2*-ll-23 22-3-683 23-17-61 22-2099 2"- 32- 59 98 2-4049 2-4099 2-32-461 2-13-17-19 2-7-607 100 2'-3*-52 23-52-41 22-52-83 2^-3-52-7 22-53-17 OF EVEN NUMBERS. 155 From 8,500 8,600 8,700 8,800 8,900 to 8,600 '8,700 8,800 8,900 9,000 2 1 2'3'13-109 2-11-17-23 2-19-229 2-33- 163 2-4451 4 2" -1063 22-32-239 2»-17 22-31-71 23-3-7-53 6 2-4253 2.13-331 2-3-1451 2-7-17-37 2-61-73 8 2^-3 -709 2^-269 22-7-311 23-3-367 22-17-131 10 2-5-23-37 2-3-5-7-41 2-5-13-67 2-5-881 2-3*-5-ll 12 26.7.19 22-2153 23-32-112 22-2203 2*- 557 14 2-32-11.43 2-59-73 2-4357 2-3-13-113 2-4457 16 2=.2129 23-3-359 22-2179 2<-19-29 22.3-743 18 2-4259 2-31-139 2-3-1453 2-4409 2-73-13 20 2'.3-5-71 22-5-431 20-5-109 22.32.5-72 93-5-293 22 2-4261 2-32-479 2-72-89 2-11-401 2-3-1487 24 2«-2131 2*-72-ll 22-3-727 23-1103 22-23-97 26 2-3-7'i-29 2-19-927 2-4363 2-3-1471 2-4463 28 2''-13-41 22-3-719 23-1091 22-2207 2*-i2.3i 30 2-5-853 2-5-863 2-32-5-97 2-5-883 2-5-19.47 32 2'-33-79 23-13-83 22-37-59 2'-3-23 22.7-11-29 34 2.17.251 2-3-1439 2-11-397 2-7-631 2-3-1489 36 2= -11. 97 22-17-127 2^^-3-7-13 22-472 23-1117 38 2.3.1423 2-7-617 2-17-257 2-32-491 2-41-109 40 2^-5-7-61 26-33-5 22-5-19-23 23-5-13-17 92-3-5-149 42 2-4271 2-29-149 2-3-31-47 9-4421 9-17-963 44 2°.3-89 22-2161 23-1093 23-3-11-67 2"- 13- 43 46 2-4273 2-3-11-131 2-4373 2-4423 2-32.7-71 48 2^-2137 23-23-47 22-3' 2*-7-79 22-2237 50 2-3^-52-19 2-52-173 2- 5"- 7 2.3-5=-59 2-52-179 52 2^-1069 22-3-7-103 2^-547 2^-2213 23-3-373 54 2-7-13-47 2-4327 2-3-1459 2-J9-933 2-112-37 56 2'-3-23-31 2^-541 22-11-199 23-33-41 22-2239 58 2-11-389 2-32-13-37 2-29-151 9-43-103 2-3-1493 60 2^-5-107 22.5-433 23-3-5-73 92-5-443 28-5-7 62 2-3-1427 2-61-71 2-13-337 9-3-7-211 2-4481 64 2' -2141 23-3-192 22-7-313 2^-277 22-33-83 66 2-4283 2-7-619 2-32-487 2-11-13-31 2-4483 68 2'-3''-7-17 22-11.197 26-137 22-3-739 23-19-59 70 2-5-857 2-3-5-172 2-5-877 2-5-887 2-3-5-13-23 72 22-2143 2^-271 22-3-17-43 23-1109 22-2943 74 2-3-1429 2-4337 2-41-107 2-32-17-29 2-7-641 76 2'- 67 22-32-241 23-1097 22-7-317 2<-3-ll-17 78 2-4289 2-4339 2-3-7-11-19 2-93-193 9-672 80 22-3.5-11-13 23-5-7-31 22-5-439 2*-3-5-37 92-5-449 82 2-7-613 2-3-1447 2-4391 2-4441 2-3«-499 84 2»-29-37 22-13-167 a*-3^-61 92-9221 23. 1123 86 2-3''-53 2-43-101 2-23-191 2-3-1481 2-4493 88 22-19-113 2^-3-181 22-133 23-11-101 22-3-7-107 90 2-5-859 2-5-11-79 2-3-5-293 2-5.7-127 2-5-29-31 92 2''-3-179 22-41-53 23.7-151 22-32.13-19 25-281 94 2-4297 2-33-7-23 2-4397 2-4447 2-3-1499 96 22-7-307 23-1087 22-3-733- 2«-139 22-13-173 98 2-3-1433 2-4349 2-53-83 2-3-1483 2-11-409 100 2»-52-43 22-3-5«-29 2^-52-11 22-52-89 23-32-53 156 PRIME FACTORS From 9,000 9,100 9,200 9,300 9,400 to 9,100 9,200 9,300 9,400 9,500 2 2-7-643 2-3-37-41 2-43-107 2-4651 2.3.1567 4 2=-2251 2'' -569 g2-3-13-59 2^-1163 22.2351 6 2-3-19-79 2-29-157 2-4603 2-32.11-47 2-4703 8 2^-563 22-32-11-23 2^-1151 22-13-179 30.3.72 10 2-5-17-53 2-5-911 2-3-5-307 2-5-72-19 2.5.941 12 2=-3-751 2'- 17- 67 22-72-47 2=-3-97 22.13-181 14 2-4507 2-3-72-31 2-17-271 2-4657 2-32-523 16 2'-7=-23 22-43-53. 310.3s 22-17-137 2'- 11- 107 18 2-3=-167 2-47-97 2-11-419 2-3-1553 2-17-277 20 2'-5-11.41 25-3-5-19 ■ 22-5-461 2;'-5-233 22-3-5-157 22 2-13-347 2-4561 . 2-3-29-53 2-59.79 2-7-673 24 2". 3-47 22-2281 2^-11-53 22.32-7-37 2«- 19-31 26 2-4513 2-3=- 132 2-7-659 2-4663 2-3-1571 28 2«-37-61 2^-7-163 22-3-769 2*-ll-53 22.2357 30 2-3-5-7-43 2-5-11-83 2-5-13-71 2-3-5-311 2.5-23.41 32 2^-1129 22-3-761 2^-577 22-2333 2''. 32. 131 34 2-4517 2-4567 2-3*-19 2-13-359 2.53.89 36 2' -3' -251 2^-571 22-2309 2''-3-389 22-V-337 38 2-4519 2-3-1523 2-31-149 2-7-23-29 2-3-112.13 40 2^-5-113 2«-5-457 23-3-5-7-11 22-5-467 2^.5.59 42 2-3-11-137 2-7-653 2-4621 2-3'-173 2.4721 44 22-7-17-19 2^-32-127 22-2311 2''- 73 22-3.787 46 2-4523 2-17-269 2-3-23-67 2-4673 2-4723 48 2^-3-13-29 22-2287 36.178 22-3-19-41 23-1181 50 2-5^-181 2- 3 -52- 61 2-5»-37 2-52-11-17 2-33-52.7 52 2=-31-73 2»-ll-13 22.32-257 2'-7-167 22-17.139 54 2-3^-503 2-23-199 2-7-661 2-3-1559 2.29.163 56 2^-283 22-3-7-109 2'-13-89 22-2339 2^.3.197 58 2-7-647 2-19-241 2-3-1543 2-4679 2.4729 60 22-3-5-151 2^-5-229 22-5-463 2<-32-5-13 22-5-11.43 62 2-23-197 2-32-509 2-11-421 2-31-151 2.3.19.83 64 23-11-103 22-29-79 2^-3-193 22-2341 23.7.132 66 2-3-1511 2-4583 2-41-113 2-3-7-223 ' 2.4733 68 2^-2267 2»-3-191 22-7-331 23-1171 22-32.263 70 •2-5-907 2-5-7-131 2-32-5-103 2-5.937 2-5-947 72 34.34.7 22-2293 2' -19 -61 22.3-11-71 2"- 37 74 2-13-349 2-3-11-139 2-4637 2.43-109 2-3-1579 76 2»-2269 2'-31-37 22.3-773 25-293 22-23-103 78 2-3-17-89 2-13-353 2-4639 2-32-521 2-7(677 80 2''-5-227 22-33-5-17 2'-5-29 22-5-7-67 23-3-5-79 82 2-19-239 2-4591 2-3-7-13-17 2-4691 2.11.431 84 2'-3-757 26.7.41 22-11-211 2'-3-17.23 22.2371 86 2-7-11-59 2-3-1531 2-4643 2-13-192 2.32-17-31 88 2'-71 22-2297 23.3»-43 22 2347 2* -593 90 2-32-5-101 2-5-919 2-5-929 2-3-5.313 2-5-13-73 92 2'-2273 2'-3-383 22-23-101 2^-587 22-3-7-113 94 2-4547 2-4597 2-3-1549 2-7-11-61 2-47-101 96 2'-3-379 22-112-19 2^-7-83 22. 3'. 29 23-1187 98 2-4549 2-32-7-73 2-4649 2-37.127 2.3.1583 100 22-52-7-13 2^-52-23 22-3-52-31 2^.52.47 22.53-19 OP EVEN NUMBERS. 157 From 9,500 9,600 9,700 9,800 9,900 to 9,600 9,700 9,800 9,900 10,000 2 2.4751 2-4801 2-32-72-11 2-132-29 2-4951 4 2s.3=.ll 32.74 23-1213 22-3-19-43 2''-619 6 2.7^.97 2-3-ieOl ' 2-23-211 2-4903 2-3-13-127 8 2^.2377 2^.1901 22-3-809 2^-613 22-2477 : 10 2.3.5-317 2.5.312 2-5-971 2.33-5-109 2-5-991 12 2^-29-41 22-33-89 2* -607 22-11-223 23-3-7-59 14 2-67-71 2-11-19-23 2-3-1619 2-7-701 2-4957 16 22.3-13-61 2^-601 22-7-347 23-3-409 22-37-67 18 2-4759 2-3-7-229 2-43-113 2-4909 2-32-19-29 20 2^-5.7-17 22-5-13-37 33-35.5 22-5-491 26-5-31 22 2-3'-23= 2-17-283 2-4861 2-3-1637 2-112-41 24 2=.2381 23-3-401 22-11-13-17 2^-307 22-3-827 26 2-11-433 2-4813 2-3-1621 2-173 2-7-709 28 2«.3-397 22-29-83 23-19 22-33-7-13 23-17-73 30 2-5-953 2-32-5-107 2-5-7-139 2-5-983 2-3-5-331 32 2«.2383 25-7-43 22-3-811 ^3-1229 22-13-191 34 2-3-7-227 2-4817 2-31-157 2-3-11-149 2-4967 36 28-149 22-3-11-73 23-1217 22-2459 »'-33-23 38 2-19-251 2-61-79 2-32-541 2-4919 2-4969 40 S'-S'-a-SS 23-5-241 22-5-487 2"- 3 -5- 41 22-5.7.71 42 2-13-369 2-3-1607 2-4871 2-7-19-37 2.3-1657 44 2^-1193 22-2411 2^-3-7-29 22-23-107 23.11.113 46 2-3-37-43 2-7-13-53 2-11-443 2-32-547 2.4973 48 22-7-11-31 2-'-32-67 22-2437 23-1231 22.3.829 50 2-52-191 2-52-193 2-3-53-13 2-52-197 2-52.199 52 2<-3-199 2= -19 -127 23-23.53 22-3-821 25-311 54 2-17-281 2-3-1609 2-4877 2-13-379 2-32-7.79 56 22-2389 23-17-71 22-32-271, 2'-7-ll 22.19-131 58 2-3^-59 2-11-439 2-7-17-41 2-3-31-53 2-13-383 60 2^-5-239 22-3-5-7-23 2=-5-61 22-5-17-29 23-3-«-83 62 2-7-683 2-4831 2-3-1627 2-4931 2-17-293 64 22-3-797 2" -151 22-2441 23-32-137 22-47-53 66 2-4783 2-33-179 2-19-257 2-,4933 2-3-11-151 68 2>-13-23 22-2417 23-3-11-37 22-2467 2*-7-89 70 2-3-5-11-29 2-5-967 2-5-977 2-3-5-7-47 2-5-997 72 22-2393 23-3-13-31 22-7-349 2'. 617 22-32-277 74 2-4787 2-7-691 2-33-181 2.4937 2-4987 76 23-32-7-19 22-41-59 2^-13-47 22.3-823 23-29-43 78 2-4789 2-3-1613 2-4889 2-11-449 2-3-1663 80 22-5-479 2^-5-112 22-3-5-163 23-5-13.19 22-5-499 82 2-3-1597 2-47-103 2-67-73 2.3*.,61 2-7.23.3i 84 2* -599 22-32-269 23-1223 22.7-353 28-3-13 86 2-4793 2-29-167 2-3-7-233 2-4943 2-4993 88 22.3-17-47 23-7-173 22-1247 2^-3-103 22-11-227 90 2-5-7-137 2-3-3-17-19 2-5-11-89 2-5-23-43 2-33-5-37 92 2'-ll-109 22-2423 28.32.17 22-2473 23-1249 94 2-32-13-41 2-37-131 2-59-83 2-3-17-97 2-19-263 96 22-2399 2^-3-101 2«'31-79 23-1237 22-3-72-17 98 2-4799 /2-13-373 2-3-23-71 2-72.101 2-4999 100 2'-3.52 2»-52.97 33.52.72 22.32.52-11 2*-5'' 158 PRIME FACTORS Prom 10,000 10,100 10,200 10,300 10,400 to 10,100 10,200 10,300 10,400 10,500 02 2-3-1667 2-5051 2-5101 2-3-17-101 2-7-743 04 2'-41-61 2»-3-421 22-2551 26-7-23 32.32. X72 06 2-5003 2-31-163 2-3«-7 2-5153 2-ll'-43 08 23-3=-139 22.7-192 2*-n-29 2=-3-859 23-1301 10 2-5-7-U-13 2-3-5-337 2-5-1021 2-5-1031 2-3-5-347 12 2'-2503 2'- 79 22-3-23-37 23-1289 2^-19-137 14 2-3-1669 2-13-389 2-5107 2-33-191 2-41-127 16 2^-313 2»-3»-281 23-1277 22-2579 2*-3-7-31 18 2-5009 2-5059 2-3-13-131 2-7-11-67 2-5209 20 2^-3-5-167 23-5-11-23 2«-5-7-73 2»-3-5-43 2^-5-521 22 2-5011 2-3-7-241 2-19-269 2-13-397 2-33-193 24 2'-7-179 2'-2531 2*-3»-71 2»-29-89 23-1303 26 2- 3= -557 2-61-83 2-5113 2-3-1721 2-13-401 28 2^-23-109 2^-3-211 2«-2557 23-1291 22-3-11-79 30 2.5-17-59 2-5-1013 2-3-5-11-31 2-5-1033 2-5-7-149 32 2«-3-ll-19 22-17-149 23-1279 22.32-7-41 26-163 34 2-29-173 2-33-563 2-7-17-43 2-5167 2-3-37-47 36 2»-13-193 23-7-181 2«-3-853 26-17-19 2«-2609 38 2-3-7-239 2-37-137 2-5119 2-3-1723 2-17-307 40 23-5-251 2»-3-5-13» 2" -5 22-5-11-47 23-3''-5-29 42 2-5021 2-11-461 2-3«-569 2-5171 2-23-227 44 22-3«-31 2^-317 22-13-197 23-3-431 22-7-373 46 2-5023 2-3-19-89 2-47-109 2-7-739 2-3-1741 48 26-157 2=-43-59 23-3-7-61 22-13-199 2^-653 50 2-3-5»-67 2-52-7-29 2-53-41 2-3»-5^-23 2-52-11-19 52 2'-7-359 23-33-47 2^-11-233 2^-647 22-3-13-67 54 2-11-457 2-5077 2-3-1709 2-31-167 2-5227 56 23-3-419 22-2539 2*- 641 22-3-863 23-1307 58 2-47-107 2-3-1693 2-23-223 2-5179 2-3»-7-83 60 2?-5-503 2^-5-127 2»-33-5-19 23-5-7-37 22-5-523 62 2-3''-13-43 2-5081 2-7-733 2-3-11-157 2-5231 64 2^-17-37 22-3-7-lP 23-1283 2«-2591 25-3-109 66 2-7-719 2-13-17-23 2-3-29-59 2-71-73 2-5233 68 2^-3-839 23-31-41 22-17-151 2'-3* 22-2617 70 2-5-19-53 2-3=-5-113 2-5-13-79 2-5-17-61 2-3-5-349 72 2'- 1259 2»-2543 2»-3-107 2'- 2593 23-7-11-17 , 74 2-3-23-73 2-5087 2-11-467 2-3-7-13-19 2-5237 76 2^-11-229 23-3-53 2»-7-367 23-1297 22-33-97 78 2-5039 2-7-727 2-3«-571 2-5189 2- 13"- 31 80 2^-3»-5-7 22-5-509 23-5-257 2»-3-5-173 2"- 5- 131 82 2-71« 2-3-1697 2-53-97 2-29-179 2-3-1747 84 2=-2521 23-19-67 22-3-857 2*-ll-59 2«-2621 86 2-3-41= 2-11-463 2-37-139 2-3»-577 2-72-107 88 23-13-97 22-32-283 2* -643 S2.72-53 23-3-19-23 90 2-5-1009 2-5-1019 2-3-5-73 2-5-1039 2-5-1049 92 2»-3-298 2<-72-13 2'-31-83 23-3-433 22-43-61 94 2-7^-103 2-3-1699 2-5l47 2-5197 2-3»-ll-53 96 2<-631 2'-2549 23-3«-ll-13 22-23-113 2S-41 98 2-33-1M7 2-5099 2-19-271 2-3-1733 2-29-181 100 2^-5'- 101 23.3-52-17 22.5=-103 2'-52-]3 22-3-53-7 OF EVEN NUMBERS. 159 From 10,500 10,600 10,700 10,800 10,900 to 10,600 10,700 10,800 10,900 11,000 02 2-59-89 2-32-19-31 2-5351 2-11-491 2-3-23-79 04 2' -13 -101 22-11-241 9«-3-993 22-37-73 23-29-47 06 2-3-17-103 2-5303 2-53-101 9-3-1801 2-7-19-41 08 22-37-71 2*-3-13-17 22-2677 93.7.193 22-33-101 10 2-5-1051 2-5-1061 2.32.5.7.17 9.5.93-47 2-5-1091 12 2'-32-73 22-7-379 23.13.103 22-3-17-53 2'-ll-31 14 2-7-751 2-3-29-61 2.11.487 9-5407 2-3-17-107 16 22-11-239 2^-1327 92.3.19.47 28-132 22-2729 18 2-3-1753 2-5309 2-23-233 2- 32- 601 2-53-103 20 2'- 5- 263 92.32-5-59 25-5-67 22-5-541 93-3-5-7-13 22 9-5261 2-47-113 9-3-1787 2-7-773 2-43-127 24 22-3-877 2'- 83 92-7-383 23.3-11.41 22-2731 26 2-19-277 2-3-7-11-23 2-31-173 2-5413 2-32-607 28 25.7-47 22-2657 23.32.149 22-2707 2''-683 30 2-3^-5-13 2-5-1063 2.5.29.37 2-3-5-192 2-5-1093 32 22-2633 2^-3-443 22.2683 a"- 677 22-3-911 34 2-23-929 2-13-409 2-3.1789 2-5417 2-7-11-71 36 2'-3-439 22-2659 2''.11.61 22-32-7-43 23-1367 38 2-11-479 2-33-197 2.7.13.59 2-5419 2-3-1823 40 22-5-17-31 2'i-5-7-19 22.3.5.179 23-5-271 22-5-547 42 2-3-7-251 2-17-313 2.41.131 2-3-13-139 2-5471 44 2^-659 92-3-887 23.17.79 22-2711 28-32-19 46 2-5273 9-5393 2.33.199 2-11-17-29 9.13.421 48 22-32-293 9'- IP 22.2687 ~ 2^-3-113 22.7.17.23 50 2-52-211 2-3-52-71 2.53.43 2-52-7-31 2.3.52.73 52 2'- 1319 2'-2663 28.3.7 22-9713 23.372 54 2-3-1759 2-7-761 2.19-283 9- 3"- 67 2-5477 56. 92-7-13-29 2^-32-37 22-2689 23-23-59 22-3-11.83 58 9-5279 2-732 2-3-11-163 2-61-89 2-5479 60 9^-3-5-11 22-5-13-41 23-5-969 22-3-5-181 2^. 5. 137 62 2-5281 2-3-1777 9-5381 2-5431 2.33.7.29 64 92-19-139 23-31-43 22-32.13.23 2''-7-97 22.2741 66 2-32-587 2-5333 2-7-769 2-3-1811 2.5483 68 2^-1321 22-3-7-127 2*- 673 22-11-13-19 23.3.457 70 2-5-7-151 2-5-11-97 2-3.5.359 2-5-1087 2.5.1097 72 22-3-881 2''-23-29 22.2693 23-32-151 22-13-211 74 2-17-311 2-32-593 2-5387 2-5437 2-3-31-59 76 2*- 661 22-17-157 23-3-449 22-2719 35.73 78 2-3-41-43 2-19-281 2-17-317 2-3.72.37 2-11.499 80 92-5-232 23-3-5-89 22.5-72-11 2' .5. .17 22. 32. 5. 61 82 2-11-13-37 2-72.109 2-32-599 2.5441 2.172.19 84 33.33,72 22-2671 2= -337 92.3-907 23.1373 86 2-67-79 2-3-13-137 2.5393 2«5443 2-3-1831 88 90 22-2647 2-3-5-353 2«.167 2.5.1069 22-3-29-31 2-5-13-83 23-1361 2-32-5-lP 22-41-67 2-5-7-157 92 94 2^-331 2-5997 22.3's.ll 2.5347 93- 19- 71 2-3-7-257 22-7-389 ,2-13-419 2^-3-229 2-23-239 22-2749 - 2-32-13.47 23.53.11 96 98 100 22-3-883 2-7-757 1 22-52-53 93.7.191 2.3-1783 22-52-107 22-2699 2.5399 2*-33-52 2''-3-227 2-5449 22-52-109 J 160 PRIME FACTORS Prom 11,000 11,100 11,200 11,300 11,400 to 11,100 11,200 11,300 11,400 11,500 02 2-5501 2-7.13-61 2-3-1867 2-5651 2-5701 04 2»-3-7-131 2»-347^ 2«-2801 93.32-157 22-2851 06 2-5503 2-3»-617 9.13-431 2-5653 2-3-1901 08 28-43 2«-2777 9' -3. 467 92-11-257 2*.23.31 10 2-3-5'367 2-5-11-101 9-5-19-59 9-3-5-13-99 9.5.7.I63 12 2'-27-53 23-3-463 9'-9803 9*-7.]01 92.32.317 14 2-5507 2-5557 9-32-7-89 2.5657 2.13.439 16 23-3'- 17 2«- 7-397 9*- 701 22.3-23.41 23.1427 18 2-7-787 2-3-17-109 9-71-79 2-5659 2.3.11.173 20 2«-5-19-29 2*-5-139 92-3-5-11-17 93-5-283 92.5.571 22 9-3-11-167 9-67-83 9-31-181 9-32-17-37 9.5711 24 2"- 13 -53 9''-33-103 93-23-61 92.19.149 25.3-7-17 26 2-37-149 2-5563 9-3-1871 2-7-809 2-29-197 28 2^-3-919 23-13-107 92-7-401 2»-3-59 22-2857 30 2-5-1103 2-3-5-7-53 2-5-1123 9-5-11-103 2-32-5-197 32 23-7-197 2»-lP-23 9"-33-13 92-2833 23.1429 34 2-3=-613 2-19-293 9-41-137 2.3.1889 2.5717 36 2^-31-89 2'-3-29 92.532 23.13.109 22.3-953 38 2-5519 2-5569 2-3-1873 2.5669 2-7-19.43 40 2^-3-5-23 2''- 5- 557 23-5-981 22.3''.5-7 2*.5.U.13 42 2-5521 2- 3"- 619 9-7-11-73 2-53-107 2.3.1907 44 2"- 11-251 23-7-199 92-3-937 2^-709 22.2861 46 2-3-7-263 2-5573 9-5623 2-3-31-61 2.59.97 48 2»-1381 2^-3-929 2^-19-37 22-2837 23. 3?. 53 50 9-52-13-17 2-5=-223 9-32.5« 2.52.227 2.52.229 52 9=-3«-307 2"- 17- 41 22-29.97 23.3.11.43 92.7.409 54 9-5527 2-3-11-13' 2-17-331 2. 7. 811 9.3.23.83 56 2^-691 2'-9789 23-3-7-67 22.17.167 28.179 58 2.3-19-97 2-7-797 2-13-433 9-32.631 2.17.337 60 2''-5-7-79 23.3«-5-31 22-5-563 95.5.71 22.3-5-191 62 2-5531 2-5581 2-3-1877 2-13.19.23 2-11-521 64 23-3-461 2'-2791 2>i'-ll 22.3.947 23-1433 66 2-11-503 2-3-1861 2-43-131 2-5683 2-32.72.13 68 2''-2767 2* -349 92.32.313 23-72-29 22. 47. 61 70 2-3'-5-41 2-5-1117 2.5.72.23 2-3-5-379 2.5-31-37 72 26-173 2=-3-7=-19 23.1409 92-9843 2*-3-239 74 2-7^-113 2-37-151 2.3. 1879 9-112-47 9-5737 76 22-3-13-71 23-11-127 22.2819 2^-32-79 22-19-151 78 2-29-191 2- 3* -23 2.5639 - 2-5689 9-3-1913 80 23-5-277 2^-5-13-43 9*.3-5-47 22-5-569 23.5.7-41 82 2-3-1847 2-5591 2-5641 2-3-7-271 2-5741 84 2'- 17- 163 2*- 3 -933 22-7-13-31 23-1423 92-32.11.99 86 2-23-241 2-7-17-47 2-33-11-19 2-5693 9-5743 88 2<-3«-7-ll 2'-2797 23-17-83 22-3-13-73 25-359 90 2-5-1109 2-3-5-373 2-5-1129 2-5.17.67 2-3-5-383 92 2^-47-59 23-1399 22.3-941 2'. 89 ■ 22-132-17 94 2-3-43« 2-29-193 2-5647 2.33.911 2-7-821 96 23-19-73 2'-32-3H 95-353 22.7-11-37 23.3.479 98 2-31-179 2-11-509 2-3-7-269 2-41-139 2.5749 100 2«-3-5»-37 28-52-7 22.52-113 23-3-52- 19 22.53-23 OP EVEN NUMBERS. 161 From 11,500 11,600 11,700 11,800 11,900 to 11,600 11,700 11,800 11,900 12,000 02 2-3^-71 2-5801 2-583t 2-3-7-281 2-11-541 04 8<-719 2^-3-967 23-7-11-19 92-13-227 2'-3-31 06 2-1 1-523 2-7-829 2-3-1951 2-5903 ' 2-5953 08 2'-3-7.137 2^-1451 22-2927 25.32.41 22-13-229 10 2-5-1151 2-3^-5-43 2-5-1171 2-5.1181 2-3-5-397 12 23-1439 2^-2903 2«-3-61 22-2953 23-1489 14 2-3-19-101 2-5807 2-5857 2-3-11-179 2-7-23-37 16 2''-2879 25-3-11= 22-29-101 23-7-211 22.32.331 18 .2-13-443 2-37-157 2-33-7-31 2-19-311 2-59-101 20 28-3=^-5 22-5-7-83 23-5-293 22-3-5-197 2«-5-149 22 2-7-823 2-3-13-149 2-5861 2-23-257 2-3-1987 24 2»-43.67 2^-1453 22-3-977 2"- 739 22-11-271 26 2-3-17-113 2-5813 2-11-13-41 2-3''-73 9-67-89 28 2^-11-131 2'-3=-17-19 2^-733 22-2957 93-3-7-71 30 2-5-1153 2-5-1163 2-3-5-17-23 2-5-7-132 2-5-1193 32 2''-3-3P 2*- 727 22-7-419 23-3-17-29 22-19-157 34 2-73-79 2-3-7-277 2-5867 2-61-97 9-33-13-17 36 2*-7-103 2^-2909 23-32-163 22-11-269 2' -373 38 2-3^-641 2-11-232 2-5869 2-3-1973 9-47-127 40 2=^-5-577 23-3-5-97 22-5.587 26-5-37 22-3-5-199 42 2-29-199 2-5821 2-3-19-103 2-31-191 2-7-853 44 2'-3-13-37 22-41-71 2^-367 22-32-7-47 23-1493 46 2-23-251 2-32-647 2-7-839 2-5923 2-3-11-181 48 2^-2887 2^-7-13 22-3-11-89 23-1481 22-29-103 50 2-3-5'-7-ll 2-52-233 2-53-47 2-3-52-79 2-52-939 52 25-19' 22-3-971 23-13-113 22-2963' 2*-32-83 54 2-53-109 2-5827 2-32-653 2-5927 2-43-139 56 2=-3'-107 2^-31-47 22-2939 2*-3-13-19 22- 72- 61 58 2-5779 2-3-29-67 2-5879 2-72-112 2-3-1993 60 23-5-172 22-5-11-53 2*-3-5-72 22-5-593 2-5-13-23 62 2-3-41-47 2-7'-17 2-5881 2-32-659 2-5981 64 2'-7^-59 2''-3« 22-17-173 23-1483 22.3-997 66 2-5783 2-19-307 2-3-37-53 2-17-349 2-31-193 68 2''-3-241 22-2917 23-1471 22-3-23-43 26-11-17 70 2-5-13-89 2-3-5-389 2-5-11-107 2-5-1187 2-32-5-7-19 72 2«-ll-263 23-1459 22-33-109 2^-7-53 22-41-73 74 2-3=^-643 2-13-449 2-7-292 2-3-1979 9-5987 76 2'- 1447 22-3-7-139 2'- 23 22-2969 23-3-499 78 2-7-827 2-5839 2-3-13-151 2-5939 2-53-113 80 2»-3-5-193 2^-5-73 22-5-19-31 23.33-5-11 22-5-599 82 2-5791 2-32-11-59 2-43-137 2-13-457 2-3-1997 84 2«-181 22-23-127 23-3-491 22-2971 2*-7-107 86 2-3-1931 2-5843 2-71-83 2-3-7-283 2-13-461 88 2'- 2897 23-3-487 22-7-421 2«-743 22-3^-37 90 2-5-19-61 2-5-7-167 2-32-5-131 2-5-29-41 2-5-11-109 92 23-3'-7-23 22-37-79 2^-11-67 22-3-991 2'- 1499 94 2-11-17-31 2-3-1949 2-5897 2-19-313 2-3-1999 96 2''- 13-223 2*- 17-43 22-3-983 93-1487 22-2999 98 9.3-1933 2-5849 2-17-347 2- 32- 661 2-7-857 100 1 2*-5«-29 22-32-52-13 23-52.59 1 22-52-7-17 95-3-53 21 162 PRIME FACTORS OF EVEN NUMBERS. Prom 12,000 12,100 12,200 12,300 12,400 to 12,100 12,200 12,300 12,400 12,500 02 a-17-353 2-3-2017 9-6101 2-6151 2-32-13-53 04 a'i-sooi 2^-17-89 2'-33-113 2*- 769 22-7-443 06 2-3»>23-29 2-6053 2-17-359 2-3-7-993 2-6203 08 2'- 19 -79 2'- 3- 1009 2*- 7 -109 22-17-181 23-3-11-47 10 2-5-1201 2-5-7-173 2-3-5-11-37 2-5-1231 2-5-17-73 12 2'-3-7-H-13 2^-757 22-43-71 93- 3"- 19 22-29-107 14 2-6007 2-3=- 673 2-31-197 9-47-131 2-3-2069 16 2^-751 2«- 13-233 93-3-509 22-3079 2'-97 18 2-3-2003 2-73-83 2-41-149 913-9053 2-7-887. '20 2=-5-601 2'-3-5-101 22-5-13-47 95.5-7-11 22-33-5.23 22 2-6011 9-11-19-29 9-32-7-97 2-61-101 9.6211 24 2= -3' -167 2'-7-433 2»-191 22-3-13-79 23.1553 26 2-7-859 2-3-43-47 2-6113 2-6163 2.3-19-109 28 2'-31-97 2= -379 22-3-1019 23-23-67 22.13-939 30 2- 3- 5- 401 2-5-1213 '9-5-1223 2-32-5-137 9.5-11-113 32 28-47 2»-3''-337 23-11-139 22-3083 9^-3-7-37 34 2- 11-547 2-6067 2-3-2039 2-7-881 2.6917 36 2''-3.17-59 23-37-41 22.7- 19-23 2^-3-257 92.3109 38 2-13-463 9-3-7-17= 9-99-911 2-31-199 2.32.691 40 2'-5-7-43 9'- 5- 607 2^-32.5-17 22.5-617 23.5.311 42 2-3'-223 2- 13- 467 2-6121 2-3-112-17 2.6221 44 2^-3011 2*-3- 11-23 22-3061 23-1543 92.3.17.61 46 2- 19-317 2-6073 2-3-13-157 2-6173 9-72.127 48 2'-3-251 2''-3037 93-1531 22-32-73 2^-389 50 2-5^-241 2-3'-5« 9-53-72 9-52-13-19 2-3-52-83 52 2'- 93- 131 93-72-31 22-3-1021 2»-193 22-11-283 54 2-3-7»-41 2-59-103 2-11-557 2-3-29-71 2-13-479 56 23-11-137 2=-3-1013 2^-383 22-3089 93-32-173 58 2-6029 2-6079 2-33-927 9-37-167 9-6929 60 2'-3'-5-67 9''-5-19 22-5-613 93-3-5-103 22-5-7-89 62 2-37-163 9-3-9097 9-6131 2-7-883 2-3-31-67 64 2^-13-29 9^-3041 93-3-7-73 22-11-281 2^-19-41 66 2-3-2011 2-7-11-79 9-6133 9-33-299 2-23-271 68 2='-7-431 2»-3='-13« 92-3067 2^-773 22-3-1039 70 9-5-17-71 2-5-1217 9-3-5-409 2.5-1237 2-5-29-43 72 2'-3-503 9''- 17- 179 9''-13-59 22-3-1031 23-1559 74 2-6037 9-3-2029 9-17-192 2-23-269 2.y-7-ll 76 9»-3019 2'-761 22-32-11-31 93-7-13-17 22-3119 78 2-3='-ll-61 2-6089 2-7-877 9-3-9063 2-17.367 80 2^-5-151 2»-3-5.7-99 93-5-307 22-5-619 2«.3-5.13 82 2-7-863 9-6091 9-3-93-89 2-41.151 2-792 84 2«- 3- 19-53 23-1523 92-37-83 2».32-43 22-3121 86 2-6043 2-3«-677 2-6143 2-11-563 9.3.9081 88 2^-1511 2»- 11-277 2>2.3 2»-19-163 23.7-223 90 2-3-5-13-31 2-5-23-53 2-5-1229 2-3-5-7-59 2.5.1949 92 2^-3023 25-3-127 22-7-439 23-1549 22.32-347 94 2-6047 2-7-13-67 2-32-683 2-6197 2-6247 96 2»-3»-7 22-3049 93-99-53 22-3-1033 2*-ll-71 98 2-23-263 2-3-19-107 9-11-13-43 2-6199 2-3-2083 100 1 S^'S'-ll' 23-5«-61 92.3-52-41 2*-52.31 22-5' OF THE ORIGIN AND CHARACTER OF MR. BRANCKER'S TABLE OF INCOMPOSITES BELOW 100,000. The following account of this table is made up from Mr. Brancker's Preface to his English Translation of Rhonius' Algebra, bearing date, White-Gate in Cheshire, April 22, 1668; and from remarks upon the table by Dr. John Wallis, in his "Discourse of Combinations, Alterna- tions and Aliquot Parts," published at London with his Treatise on Alge- bra in 1685. The Table, the Preface, and the Discourse are contained in a volume of Mathematical Tracts published by Francis Maseres, Esq. cursitor baron of the Court of Exchequer, in London in 1795; a copy of which is to be found in the Boston Athenaeum. In 1669 one Rhonius published at Zurich in Switzerland, a work upon Algebra, in the German language, in a quarto volume, to which there was appended a Table of Composite and Prime Numbers, covering six leaves of the book. In 1662, " a good friend of Brancker handed him a copy of that work and told him, he much desired to read it in some language that he under- stood." Brancker then promised, as soon as his leisure would permit, to translate it into English; which he afterwards did. When Brancker first sent his translation to the press, he gave an order that the table should be re-printed precisely as it stood in the original of Rhonius. But afterwards, he heard there was in London a person of note, (Dr. John Pell,) worthy to be made acquainted with his design. Being per- mitted to speak with him, he found him not only able, but willing to assist him. He shewed Brancker the way of making the Table, of exam- ining it, and of continuing it; and encouraged him to extend it to 100,000. This was accordingly done; and the Table thus extended was published in London as an Appendix to the Translation in 1668. As a matter of curiosity, and in respect to its author or authors, the title page is herein re-printed at large. Its running title, in the re-print by Maseres, is "Mr. Thomas Brancker's Table of Incomposit or Prime Numbers less than 100,000." It contains however all the composite as well as the prime numbers ending in 1, 3, 7 or 9 within its limits, and the least one of the prime factors of each of the former. Every prime number in that re-print is designated by the letter {p,) and every factor which is the square root of its composite by a dash or black line drawn under or over it. In the present work blank spaces or dots are substituted for (p,) while the dash is preserved, but it is always placed under the number which is the root, and this is in a smaller type than the other figures. The number of primes in each co- 163 164 ORIGIN OF BRANCKER'S TABLE. lumn or century has been ascertained and expressed by the author of this book. The following are some of the remarks made upon this table by Dr. Wallis in his above mentioned Discourse : He says, "we have at the end of Dr. Pell's [Rhonius'] Algebra, trans- lated and published ^y Thomas Brancker, in the year 1668, a compen- dious table of all odd numbers, not ending in 5, as far as 100,000; shew- ing not only which of them are prime numbers, but also by what smallest prime nurpber every other of them may be divided." In regard to the character of the table for accuracy, he says, "Now because in such tables, it is of great moment that they be carefully com- puted and exactly printed, (because mistakes therein are not easily ob- served and corrected by the reader's eye,) I have taken care to examine that whole table very exactly in the same method and with the same pains as if I were to compute it anew, and I find that though it had been computed and printed with great care, some few mistakes and but a few, have escaped the corrector's eye; most of which are noted in the table of errata printed with it. Besides, which I have noted these that follow, which to save another reader the like labor, I have thought fit for his ease and satisfaction here to note; and these being also amended as here directed, besides those noted in the printed errata, the fable will then be very accurate and I think without any error." Then follows Dr. Wallis' table of errata discovered by him. As Mas6res' book contains both tables of errata, it would have been easy to make all the corrections indicated by them, had we an original copy of Brancker's table, which was revised by Dr. Wallis ; and then we might have relied upon its accuracy from his examination. But although nearly all the proper corrections are made by Maseres, yet some other errors having been found in his copy, the author of this present book has performed a similar labor with that of Dr. Wallis, and with like care and pains, in order that the public may have the utmost confidence in the entire accuracy of that portion of the table herein contained. All the necessary corrections having been made after the stereotype plates were cast, it is scarcely possible that a single error remains. EDWARD HINKLET. Baltimore, Sept. 1, 1852. fPART OP A TABLE ENTITLED] AN APPENDIX ENGLISH TRANSLATION RHOMUS'S GERMAN TREATISE OF ALGEBRA, MR. THOMAS BRANCKER, M. A. ^ AND FCBLISHED BT HIM, "WITH THE ADVICE AND ASSISTANCE OF DR. JOHN PELL, AT LONDON, IN THE YEAR 1668; CONTAmiNe A TABLE OF ODD NUMBERS LESS THAN ONE HUNDRED THOUSAND, PIRSTj WHICH or THEM ARE INOOUPOBITE OR PRIME NUMBERS, AND SECONDLY, THE FACTORS OR CO-BFFZCIEHTS, BT THE MO LTIP LI CATION OP WHICH THE OTHERS ARE PRODUCED, SUPFUTATED, OR COMPUTED, BT THE SAME THOMAS BRANCKER. 166 BRANCKER'S TABLE OF COMPOSITE AND 200 201 202S03 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 01 03 07 09 3. 83 3. 11 23 3 13 127 3 3 7 3 127 3 137 3 17 3 113 11 3 19 3 H 13 17 19 11 . 3 17 3 137 17 7 3 13 3 109 3. 43 3 7 ,101 13 17, 3. 37 17 7 23 21. 23. 27 29. 73 3 1113 3 29 3 13 3 31 17 41 19. is 31 3 7 3 11 3 7 3, 43 139 13 83 3 11 3 31 33 37 39 3 13 3 29 3 107 3 3 47 3 37 83 67 7 3 109 3 3 17 3 67 31 103 41 43 47 49 11 3. 31 3. 3. 11. 3. 3 19 3 53 11 . 13 7. 11. 3 37 3 41 3 89 17 23 37 17 47 51 S3 57 59 47 3 113 3 41 107 19. 7 73. 29 3 3 131 3 13 3. 59 3. 11 3 29 3 7 61 63. 67 69 29 isi' 67 13 3 19 3 3 n 3 47 7, ii 3 11 3 71 73 77 79 23 17 19 13 7 107 3 109 3 47 13 63 7 3 131 3 127 7 si 81 83 87 89 17 3 89 11, 19 13 137 17 31 1139 3 29 3 13 '7 61 7 113. 23 3 3. 23 3 13 3 11 91 93 97 99 3 71 3 101 103 7. '53. 31 3 il03 3 59 43' 17 3 7 3 3 7 3 11 3, 107 3, 19 109 3 13 3 13 12 10 12 11 13 11 10 13 11 11 INCOMPOSITE NUMBERS LESS THAN 100,000. 167 220 221 01 03 07 09 11 13 17 19 21 23 27 29 31. 33 37 39. 41 43 47 49 51- 53 57 59 61 63 67 69 81 83 87 89 13 29 71 73- 77 79- 71 3 13 3 91. 93. 97 99 11. 17 19 7 225 149 3 53 3 113 3 7 3 12 9 323 29 137 23 7 89 13 224 3 43 3 113 43 3 101 3 13 9 8 225 226 97 7 13 23 3 139 3 17 131 19 11 12 3 127 3 227 3 73 3 31 11 13 228 151 3 137 229 '37 si 3 101 3 7 103 7 3 127 3 83 is 109 230 16 231 13 3 7 3 232 23 3 139 3 7 233 3 7 3 11 83 41 103 67 7 19 3 149 3 234 7 3 11 131 235 71 19 11 101 3 103 3 236 13 237 137 3 151 3 11 238239 13 7 29. 11 31 3 113 3 7 17 3 107 7 29 11. 3 7 23 13 3 103 168 BRANCKER'S TABLE OF COMPOSITE AND I 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 01 03 07 09 H 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 11 59 101 13 53 7 149 107 12 7 19 3 109 3 19 3 107 3 3... 7 13 3 137 3 13 3 10 19 3 109 3 11 17 13 137 97 107 3 11 3 109 3 13 131 3 29 3 13 11 139 3 7 3 113 11 17 19 151 11 43 37 127 7 17 19 73 7 13 3 3 . 101 3 3 17 3 7 3 67 109 11 12 29 3 7 3 11 11 13... 7 59 29 31 3 17 83 131 31 17 3 53 7 11 3 71 157 3 11 3 13 113 3 23 47 131 3 13 3 149 3 T 19 107 17 43 3 103 3 19 41 113 7 37 3 101 3 17 7 3 11 19... 3... 10 81 12 INCOMPOSITE NUMBERS LESS THAN 100,000. 169 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 960 3 "3 31 261 43 3 262 7 73 263 264 265 11 3 157 41. 43 47 , 49 51 53 57 59 109 17 ^ 37 59 ... 19 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99, 67 3 131 11 17 3 29 11 7 113 13 3 19 7 3 43 3 137 47 109 61 41 266 267 268 269 43 13 7 3 17 137 3 31 11 53 3 139 3 47 3 '3. 3 139 3 270 149 3 7 3 13 11 3 107 3 13 13 3 113 3 41 271 3 61 3 151 41 272 273 37163 7 19 73 3 59 3 149 11 137 274 275 276277 3 11 79 3 157 23 7 101 11 151 97 3 13' 3 11 13 3 17 23 a 3 7 3 103 3 139 3 61 23 97 3 29 . 13 103 11 19 278 279 11 37 19 31 107 43 19 131 43 43 17 13 3 11 3 13 103 3139 17 3 7 3 11 17 3 7 3 19 3 13 73 13 83 11 101 7 8j 11 11 10 10 8 10 11 12 10 10 8 10 5 10 ~^2 9| 14 9 11 170 BRANCKER'S TABLE OF COMPOSITE AND 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 2981299 3 109 3 157 11 7 3... 23 3 157 3 127 3 13 3 79 113 19 11 11 19 17 37 17 3 101 3 11 109 13 7 ii 12 14 83... 3 7 137 47 3. 13 3 7 23 149 3 19 3 127 167 3 11 3 3 67 29 3 11 113 3 31 3 13 127 17 19 19 3 131 43 7 11 37 61 7 151 103 31 163 19 3 109 7 139 3 3 7 149 31 11 11 11 163 19 23 11 109 127 101 17 10 3113 3 107 149 13 47 7 11 3 7 3 13 3 131 3 7 151 71 3 17 3 11 23 53 3 11 3 19 13 17 3 31 3 157 71 167 7 29 10 3 131 INCOMPOSITE NUMBERS LESS THAN 100,000. 171 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 01 03 07 09 31 "7 U57 3. 3. 17 71 3 127 3 29 7 101 11 113 23 13i 3 7 3 37 19 61 17 ai 13 17 19 17 "7 11 3 53 3 29 3 3 173 3 101. 3, 43 3 101 3 7 3 7 3 59 21 23 27 29 7 3 47 3 47 167 " 19 3. 13 23 131 7 3 113 3 109 3 17 3 157 3. 11 13 7 11 53 103. 3. 137 3 7 3 31 33 37 39 29 3 7 3 11 3. 19 3. 79 73 7 59 3. 163 3. 17 3 17 3 149 47 7 17 29 37 11 109 19 41 43 47 49 11 13 isi 19 ii 13 'i9 3 109 3 7 157 23. 3. 13. 3 3 17 3 43 51 53 57 59 13 79 3 127 3 7 137 3 7 3. 23 59 3 13 3 83 )107 3 3 139 3 11 113 11 7 89 3 61 63 67 69 23 3 107 3 3 53 3 83 41 19 ii. 29. 7 3 173 3 47 3 11 3 71 37 3. 11. 3 23 3 151 3 11 3 31 is 7 71 73 77 79 3 11 3 103 3 31 3. 29 37. ii 47 is 11 3 137 3 31. 131 "7 23 127 71 3 113 81 83 87 89 3. 67 3. 107 11 31 7 11 3 43 3 3 61 3 13 3 7 3. 67 13 3 "3 11 3 29 3 91. 93 97. 99 il09 is 41 3. tll3. 9 12 9 47 3 3. 37 41 7 13 19 17 3 139 3 13 8 17, 11 137 10 13. 3. 12 11 17 12 3 7 3 13 7 3 19 3 10 10 3 "3 10 167 13 7 11 7 6 172 BRANCKER'S TABLE OF COMPOSITE AND 320 321 322 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 163 17. 101. 3. 3 103 3 7 179 , 3 73 3 51- 53 57- 59- 61 63 ■ 67 69- 71 73 77 79 81 83 87 89 91 93 97 99 11 31 3 7 3 19 167 103 3. 11 3. 29 3 19 3 53 23 7 323 324 11 7 ii" 13 325 7 7 3 163 3 3...i 19 59 3| 7 8... 47 13 3,. 19 31 13' 11 83 1391 3 3 7 53 7 3. 43 29! 3. 179 3 I 13! 11 19 13 29 326 327 103 17 11 97 328 329 3 107 lOl 8 330 13 11 331 332 79 7 lis 101 10 333 3 167 3 43 7 3 107 3 334 335 127 ii 109 7 336 11 12 337 67 37 13 19 41 3 131 3 338339 11 13 107 31 149. 3. 23 3. 13 43. 3 11 3 97 7 19. 3. 11 3 19 3 17 31 7 3. 109 10 INCOMPOSITE NUMBERS LESS THAN 100,000. 173 340 341342 343 344 345 346347 348 J_ 349 350 351 352 353354355 356 357 358 359 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 101 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 149 13 23127 11 47 173 3 91 73 93 97 99 103 7 13 11 19 43 isi 7 7 3 13 3 127 3 13 179 47 3 3 131 109 17 3 53 7 3163 11 41 11 17 29 17 53 103 3 149 3 17 11 3 17 13 37 181 7 71 151 29 113 13 11 10 10 11 157 3 19 3 13 181 7 43 13 10 23 13 131 13 11 7 11 167 131 11 113 7 3. 3. 3 13 3 101 3 17 41 3 7 13 3 151 3 7 37 179 is 10 13 3 29 23 3 13 3 157 61 3 7 3 149 113 3 11 19 3 13 , 3. 103 31 7 101 181 13 97 13 29 17 37 19 3 83 3127 73 3 11 103 3 157 41 7 13 13 3 11 '"7 17 19 11. 4 7 12 10 J 74 BRANCKER'S TABLE OF COMPOSITE AND 360 361 362 363 364 365 366 367 368369 370 371 372 373 374 375 376 377 378 379 01 03 07. 09 13 79 .163 23 7 3. 113 3, 19 31 "ii 103 3 7 3 151 29 i67 11 13 17 19 181 3 7 3. 19 11 '23 131 3 11 3 127 11. 3 67 11 3 17 3 3 29 3 43, 3. 13 59 3 31 3 7 21 23 27 29 3 13 3 7 29 11 17 59 23. 7 13. 137 107 3 163 3 23 13 7 3 157 3 17 3 191 3 3 109 3 11 31 33 37 39 137 3 23 n 47 3 7 3 3 7 3. 61 23 109, 17 19 29 7 31 3 23 3 11 'ei 7 3 157 3 83 7 59 11 41 43 47 49 23 7 11 13 19 163 3. 11 3 ■ 7 11 13 67 17 11. 167 "7 193 3. 107 3. 13 7 U. 3 19 3 137 51 S3 57 59 3 31 3 107 7 13 101 3 103 11. 139 43 137 '29 3 7 3. 13 41 3 17 13 7 47 23 3 61- 63 67. 69 13 3 41 3 61 3 37 3 3 191 3 7 13 101 19 7, 3. 83 3, 11 3 11 3 179 43 71 73 77 79 43 109 3 61 3 11 37 ii 7. 43. 3. 11 3. 11 .103 3. 131 3 7 13 11 , 7. 3. n 3. 53 3 101 3 41 107. 3 37 3 3 13 3 163 81 83 87 89 3. 151 7 13. 131 11. m 3 11 3 157 7 13 7 "37 3 23 3 7 37 "ig 7, 3. 13 3 3 43 3 91 93 97 99 10 3 "3 10 151 3 17 3 3, 23 3. 17. 11 31 7 12 3 79 3 11 29 7 '23. 10 139 61. 149 12 8 15 11 10 13 10 INCOMPOSITE NUMBERS LESS THAN 100,000. 175' 380 381 01 03 07 09 3 7 3 191 389 383 384 11. 13 17 19 21 23 27 29 31 33 37 39 193 47 11 17 13 385 11 3 193 41109 43 47. 49 43 7 37 51 S3 57 S9 61 63 67 69 71 73 77 79 13 19 7 3 167 3 23 19 71 107 41 103 139 7 97 386 387 388 7 11 3, 13 59| 19 3 7 59. 81 83 87 89 113 "7" 41. 3 101 91 93 97 99 181 3 7 131 23 13 19 19. 3 151 197 31 17 79 109 7 61 3 J 37 3 17 3 173 47 101 11 7 167 13 389 390 391 71 11 392 61 393 137 7 17 13 47 11 163 23 3 103 3 17 109 3 197 3 113 19 394 395 31 7 157 17 7 13 127 3 7 3 139 3 149 3 3 7 3 107 173 37 3 139 3 7 47 113 396397 11 7 173 3 11 29 3 7 23 3 103 11 163 17 101 11 3 151 3 398399 17 73 127 17 67 79 7. 61 107 167 179 11 3 73 127 3 83 3 13 19 113 10 10 6 7 12 11 9 10 12 9 12 7; 11 12 8 176 BRANCKER'S TABLE OP COMPOSITE AND 400 401 402 403 404 405406 407 408 409 410 411 412 413 414 415 416 417 418 419 3 01 03 07 09 13 109 11 11 13 . 17 19 21 23 27 29 31 3 13 3 53 79 isi 37 31- 33 37- 39- 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 3. 67 3. 11 11 3 41 3 103 17, 3 11 3 13 149 3 7 167 11 3 127 23 11. 7 47 loi 3 61 11 83 101 3 107 3 47 3 113 3 13 3 109 3 43 193 139 13 131 3 7 3 37 11 107 3 17 3 3 131 3 23 13 23 3 47 101 3 3109 3 79 3 7 13 3 149 3 7 157 7 61 13 3 173 3 29 29 "7 181 3 '3 '3 '3 107 7 11 3 179 3 17. 97 3 167 . 3 13 89 197 11 13 3 151 3 59 11 17 7 7 13. 109 83. 37 41, 3 7 3. 11 11 7 23 3 19 3 17 41 7 149 3 13 3 3 29 3 19 3 13 3 11 U99 163 11. 11 12 INCOMPOSITE NUMBERS LESS THAN 100,000. 177 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 01 03 07 m 97 3 7 3 13 17 3 7 3 7 3 109 3 19 3 13 3 137 3 "7 3 23 3 13 "3 107 3 7 3 "3 "k 19 13 11 7 3 "3 83 41 3 139 3 59 7 3 11 3 109 '3 71 3 11 43 23 19 29 41 3 11 3 11 13 17 19 43 3 23 3 7 ]3 3 7 3 29 17 11 101 3 7 3 13 7 3 17 3 "43 19 17 3 11 3 31 3 47 3 11 13 7 167 3 3 19 3 "3 7 79 23 11 3 "3 "3 11 3 13 53 "7 3 "3 53 "3 '3 193 7 43 29 s. 3 ■3 37 21 23 27 29 31 33 37 39 41 43 47 49 3 "3 13 73 3 103 3 ... 3 59 3 7 101 13 23 71 3 7 3 47 7 3 "'3 ii 113 3 "3 11 3 17 3 13 29 7 17 3 "3 13P "3 37 3 7 173 137 3 71 3 19 181 3 '3 '23 73 7 3 13 4? 167 3 13 3 Ij 3 7 11 3 127 3 17 "19 7 "7 29 3 157 3 "3 "3 13 7 17 151 3 89 3 7 3 ... 13 151 '79 3 3 7 3 "3 7 3 7 3 "3 23 '67 29 37 23 i93 ? 3 3 "3 179 ' 7 3 13 3 17 3 "3 11 83 59 61 'i7 7 19 3 89 3 67 3 'I 11 101 3 13 3 7 ii 17 3 19 3 3 101 3 191 17 3 11 3 53 3 59 3 7 17 163 13 197 *53 7 3 '3 71 'si 3 "3 11 3 7 19 3 157 3 3 17 3 113 53 3 83 3 3 23 3 "7 11 11 3 7 51 53 57 59 3 11 3 137 61 3 11 29 3 41 3 "3 "3 17 7 "ii 3 13 3 29 "3 11 3 ■73 3 3 7 3 181 7 3 191 3 'ig 'is 3 97 3 43 "3 149 3 67 '7 3 "3 61 "3 113 3 3 7 3 ii 103 17 3 7 3 7 61 63 67 69 "3 23 3 7 11 149 3 13 3 43 11 3 13 3 "7 3 31 3 37 3 "3 61 7 'i9 3 "3 163 "3 17 3 17 3 7 "3 7 3 131 103 17 31 3 7 3 17 7 3 19 3 '47 13 3 107 3 11 23 3 "3 "7 3 13 71 73 77 79 "7 29 3 181 3 41 3 67 3 7 3 "3 71 139 3 "3 11 43 3 53 3 97 7 11 3 19 3 23 23 3 "3 i09 13 113 3 11 3 7 29 3 7 3 11 3 7 3 31 7 3 '3 19 73 17 11 3 '3 13 31 3 107 3 7 81 83 87 89 91 93 97 99 3 "3 3 11 3 19 23 3 3 7 97 37 3 "'3 179 3 "3 137 19 13 7 3 53 3 67 3 11 s 29 7 19 3 "3 73 13 3 43 3 "ii 157 3 41 3 7 11 3 7 3 ... 3 7 3 7 3 "'.3 3 "3 31 "7 13 3 7 3 11 3 3 191 3 41 11 3 7 3 59 3 13 3 19 3 41 'ii 7 3 47 3 13 3 23 3 3 "3 7 "3 7 3 "29 '23 3 "3 11 7 3 29 3 7 3 "3 13 37 89 'ig 3 3 127 10 10 10 10 15 7 9 13 8 10 9 7 8 7 9 8 10 11 5| 11 23 178 BRANCKER'S TABLE OF COMPOSITE AND 440 441 4431443 444 445 446 447 448 449 450 451 452 453 454 455 456457 4584 01 03 07 09 11 13 17 19 3. 79 3 7 21. 23 27. 29. 31 33 37 39 41 43 47 49 51 53 57 59 3 11 3 47 13 61 63 67 69 3 139 3 127 71 73 77 79 81 83 87 89 17 13 91 93 7 97 3 99 11 11 3 151 3 3 7 3 59 7 13 3... 193 3 10 10 13 43 3 101 11 41 13 17 11 29 167 m 127 3 107 13 10 79 11 31 11 11 73 31 107 19 13 10 29 3 103 3 89 113 83 3 43 3 103 11 7 29 137 101 47 67 23 173 13 3 37 3191 11 29 199 3 7 3 10 8| 10 109 17 3 17 3 131 3] 163 3 19 61 3 67 37 23 19 3. 13 7 11. 11 17 109. INCOMPOSITE NUMBERS LESS THAI^ 100,000. 179 460 461 469463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 01 03 07 09 157 179, 13 139 11 13 17 19 13 3 107 3 21- 23 27- 29 17 7 .193 )163 31 33 37 39 191 13 19 7 11 37 113 19 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 3, 41 3. 13 131 103 7 31 11 149 3 151 3 7 71 199 11 193 181 11 13 149 10 97 3. 31 10 101 19 83 13 7 3 167 3 43 31 3139 79 11 13 59 31 131 211 151 7 67 13 17 103 3 179 3 23 197 17 41 3 109 3 10 97 19 7 3 113 3 127 3 11 3 3 103 3 13 199 13 3 113 3 7 12 17 13 7 , 3 23 3 , 137 3. 173 17 11 7 11 . 1191 3 3. 59 3 17 3 163 109 3 79 7 199 11 23. 151 23 11 3 7 3. 13 3, 71 3 7 83 47 .211 19 11 13 47 37 3 11 3 7 11 180 BRANCKER'S TABLE OF COMPOSITE AND 480 481 482 483 01 03 07 09 H 13 17 19 103 11 73 41 7 31 , 19 211 21 23. 27 29. 3 17 3 31 33 37 39 41 43 47 49 127 37 7 7 107 23 51 53 57 59 61 63 67 69 71 73 77 79 179 3 13 3 71 53 131 484 485 139 3 7 3 11 59 3 193 3 3137 17 3 13 7 101 81 83 87 89 3 7 3. 19 53 109 43 91. 93 97 99 11 157 10 11 486 7 3 71 11 7 8 12 13 17 31 487 488 31 113 53 67 13 19 7 3 . . 181 11 489 3 7 3 113 167 3 7 3 37 10 490 109 17 19 31 491 19 181 3 492 493 157 3 7 3 11 3 211 3 494 3 "3 13 19 29 3 11 3 149 103 13 23 11 3 59 127 3 31 3 13 495 496497 97 11 61 47 3 7 9 9 43 193 iis 7 107 13 179 3 17 101 3 498 31 53 13 11 499 139 7 11 29 109 31 7 3 13 3... 11 17 7 is 10 3 19 3 7 13 11 "7 79 3 199 13 11. 17 3, 73 11 3 157 47 71. 53 31 67 47 17 29 107 3 7 3 23 151 3 7 3 3. 41 3 10 17 10 INCOMPOSITE NUMBERS LESS THAN 100,000. 181 500 501 502503 504 505 506 507 508509 510 511 512:513 514 515 516 517 518 519 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 3. 31 3 43 17 3 61 11 .. 3 23 7 149 3 13 3 381 3 41 43 47 49 51 53 57 59 163 3 11 7 113. 191 11 7 61 63 67 69 71 73 77- 79 19 81 83 87 89 61 11 13 3 7 3 31 91 93. 97 99 53 17 137 19 13 3179 3 211 11 3 109 3 10 10 9 83 41 101 10 223 23 103 3 37 197 97 3 113 3 37 101 23 11 3 109 3 11 29 13 127 29 139 17 163 3 211 3 3 193 3.. 19 7 12 181 19 17 3 13.. 163 3 17 3 151 3 43 71 223 11 3. 5 8 9 9 19 53 107 13 167 10 19 101 3 3 19 11 47 3 7 83 191 17 3 103 13 149 29 7 3. 103 17 .3 197 3 23 193 7 113 43 13 47 3 139 3 3 19 8 14 19 13 11 10 3 137 3 11 3 167 3 127 7 3 11 3 223 7 3 157 3 59 3 227 3 7 3 11 3 182 BRANCKER'S TABLE OF COMPOSITE AND 520 52l!5a2|523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 3 107 91 13 93113 97 59 99 53 1193 19 17 109 7 11 79 47, 11 3 113 19 7 103 13 11. 3. 167 13 61 23 23 10, 9 3 179 3 11 137 3 97 3 3 7 3 47 131 3 107 3 139 3 19 3 7 3 149 3 23 151 S, 8 113 3 89 3 11 13 227 3 13 3 211 11 173 11 139 11 7 137 223 3 151 3 71 17 s 7 3 11 31 229 3 107 3 67 193 3 127 3 149 3 61 3 73 3 59 3 109 3 13 3 131 13 23 3 103 3 7 191 3 13 3 9 12 61 223 71 59 17 11 11 173 13 7 107 3 19 3 11 199 163 79 INCOMPOSITE NUMBERS LESS THAN lOO.i ,000. 183 540 541 01 03 07 09 11 13 17 19 21 23 27 29 53 13 31 33 37 39 41 43 47 49 3 29 3 173 542 67 151 59 13 211 7 543 51 53 57 59 191 3 61 63 67 69 71 73 77 79 139 23 17 41 81 83- 87 89 31 41 19 11 3 17 3 17 227 7 29 91 93 97 99 47 ii' 83 544 545 546 547 19 11 227 193 3. 3. 101 53 239 3 7 548 3 7 3 23 549 550 59 43 3 7 3 13 73 109 53 3 31 3 11 11 il07 7 47 3 3. 197 11 13 3 19 3 157 23 3 7 3 137 109 3 7 3 7. 3, 23 3 3 149 3 17 29 3 7 3 71 13 11 9 10 23 7 3. 37 3 179 3 71 7 131 19 551 552553 13 554 157 19 43 11. 555 67 37 3 101 7 139 n 229 3 197 3 10 3 211 3 19 556 557 558 559 17 53 3. 17 41. 11. 3 181 3 23 197 127 13 11 179 3 233 3 43 3 17 3 11 7 '47. 37 7 3 11 3 .199 103 7 23 31. 3. 3 , 139 7 3 107 3 7 13 107 191 97 3 223 3 7 12 13 7 29 10 184 BRANCKER'S TABLE OF COMPOSITE AND 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 01 03. «7 09. 43 7 11 3. 23 3 7 79 43 7 3 109 3 3 131 7 19. 13 3 79 3 11 13 17 19 79 3 13 3 3 199 3 19 7 11. 43 13 47 11 23 19. 223 37 13 31 53 17 . 7 157. 13 3 17 3 7 29 11 21 23 27 29 7 11 179 43 17 157 23 7 29 3 3. 131 3. 17 13 3 127 3 7 89 151. 97 23 1197 3 3. 11 3 53 31 33 37 39 3, 137 3 7 53. 3 53 17 7 11 113 13 3 3. 11 3 7 3 7 3. 163 13 ii. 3 151 3 41 43 47 49 103 3 29 3 3 193 23 179. 3 11. 3 3 7 3. 17 13 fl67 51 53 57 59 3 233 3 89 13 3 101 3 7 181 53 3 19 3 211 139 3 13 3 59 3 3 83 3 41 17 '47 61 63 67 69 127 3 157 3 163 13 7 7 101 19 3 29 43 3 149 3 3 173 3 37 7 !i6i 149 3 7 3 71 73 77 79 47, 3 7 3, 3 7 3. 167 149 23 3 227 3 11 ,103 3 13 3 229 101 7 13'; 11 3 31 3 29 37 81 83 87 89 17 11 13 iis 17. 7 3 71 3 11 3 163 3 19 211 3 13 3 59 . 71 3. 37 3. 7 107 13 3 23 3 103 91 93. 97 99. 181 41. 19 17 3 7 3 11 17 3 7 3 31 12 7 3 13 3 10 17. 11 3 "3, 13 37. 3. 11 3 23 3 11 10 29 3. 3. 239 10 31 , 3. 29 7 . 12 3 11 3 3 59 3 INCOMPOSITE NUMBERS LESS THAN 100,000. 185 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 01 ,03 07 09 11 13 17 19 ^1 23 27 29 31 33 37 39 17 137 7 199 7 3 13 11 37 41 43 47 49 SI S3 S7 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 10 7 163 139 3 43 3 107 227 139 "7 31 3 211 3 17 19 101 13... 3 167 3 7 n "IT 8 127 3 7 13 23 103 11 131 3 151 3 13 3 223 157 3 233 11 23 67 3 229 3 71 31 17 13 31 67 17 3 137 3 3 31 3 127 167 3 19 3 17 113 7 97 73 3 149 3 191 3 137 3 11 41 79 193 3 11 3 103 3 7 3 3 67 3 3 109... 3 7 3 . 13 11 37 3 43 101 11 11. 10 41 12 3. 113 12 10 227 3 67 13 109 29 211 3 13 3 11 41 211 13 7 19 Ifl 3 17 3 3... 37 3 10 11 11 71 11 181 3 11 3 163 3 29 11 149 3 73 3 13 11 3 11 3 3 23 7 3 151 3 167 17 31 3 61 131 19 3 191 3 17 233 12 10 3 37 3 7 223 239 3 17 3 u 186 BRANCKER'S TABLE OP COMPOSITE AND 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 01 03 07 09 11 13 17 19 29 . 3. 23 . 3 47 21 23 27 29 31 33 37 39 3 193 3 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83. 87 89- 91 93 97 99 229 13 3 179 3 23 3 7 3 19 17 3 7 3 II 13 73 17 11 3 1139 3 7 223 13 19 83 73 103 13 11 17 3 7 3 197 13 3 191 3 13 3 131 3 11 131241 3... 9 8 17 23 107 139 3 149 59 11 101 37 3 7 3 17 7 227 79 173 3 197 3 7 71 29 233 83 43 17 239 19 47 109 7 137 227 19 229 7 13 23 3 19 3 7 103 31 3 101 3 11 3 211 3 17 3 7 3, 107 17 . 24i 23 37 151 13 23 127. 3 11 3 7 41 3 7 3 223 3 163 3 U si 3 29 3 17 10 3, 19 3. 199 7 17 103 11 13 47 13 7 10 INCOMPOSITE NUMBERS -LESS THAN 100,000. 187 620 621 622 623 624 625 626 627 628639 630 631 632 633 634635 636 637 638 639 01 03 07 09 11. 13 17. 19 7 )179 11 21 23 27 29 109 13 7 11 31 33 37 39 41 43 47 49 51 S3 57 59 229 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 23 139 13 3. 17 3. 19 11 11 13 157 3 11 3 101 103 3 31 3 3 137 10 47 97179 3 7 3 61 11 3 199 3 167 7 13 .43 11 7| 9 3 149 17 131 71 31 3 11 3 3 223 3 29 3 233 97 3 239 3 41 227 19 11 3 a5i 3 113 3 19 3 79 71 89 ii 223 3 233 3 17 19 59 11 137 11 13 23 137 229 11 11 17 127 3 199 3 13 3 7 3. 73 111 7 3 167 3 139 17 107 3 23 3 17 103 7 13 6J 3 241 3 61 13 173 '7 11 151 3 13 23 3 227 3 131 19 83 29 3 "3 79 "7 41 3 97 3 101 17 13 . 37 3 103 3 67 19 7 3 17 3 43 11 13 3 31 3 7 13 167 3 47 3 23 17 3 11 3 137 3 109 3 61 3 181 3. 11 11 1188 BRANCKER'S TABLE OF COMPOSITE AND 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 01 03 07 09 11 3 13 3 107 7 53 3 251 3 11 3 229 3 41 . 47 13 113 iai 61. 17 31 13 109 3. 17 3. 3 59 3 17 11 13. 17 19. 7 157 149, 3 73 3 31 149' 7 1163 3 3 139 3 241 3 7 3 149 11 23 19 3 29 3 21 23 27 29 73 3 43 3 3. 11 il31 3 3. 113 3, 173 3. 11 3, 241 11 7. 83 "ii 211 137 29 11 31 33 37 39 11 17 3- 59 3 ■ 31 23 "7' 11. 29 11. 3 79 3 223 19 13. 43 3 7 3 233 41 43 47 49 227 17 41 47 3 37 3 229 233 19 7 17 3 127 3 13 101 3 3 101 3 107 193 3 29 3 13 19 3 101 3 31 7. 3. 11 41 3 13 29. 11 37 23 3 7 3 51 S3 57 59 13 3 7 3 83 7. 3. 139 3 73 13 7 31 3 17 3 23 3 11 3. 29 3. 67 11 3 47 3 19 101 ii 61 63 67 69 29 79 179 3 7 3 n 13 191 59 3 7 3. 23 11 3 239 13 167 7 3. 31 3 163 3 131 53 i73 7 67 7. 199 3 41 71 73 77 79 3 17 3 139 11 . 17 13 31 7 3 211 3 29 3 43 3 181 3. 59 3. 3 13 3 29 7 233 41 17 37 3 17 3 81 83 87 89 13 7 , 3 53 151 37 11 151 7 23 3, 11 3. 43 7 19 "is, 3, 157 3 19 7 91 93 97 99 107 11 7 239 3 113 3 11 'si 23 11 17 103 ii . 8 8 10 7 3 11 3 12 109 "li 13 3, 17 3 179 3 12 10 7 131 13 3 31 10 INCOMPOSITE NUMBERS LESS THAN 100,000. 189 660 661 01 03 07 09 13 3. 149. 11 11 13 17 19 21 23 27 29 251 7 107 3 103 3 31 33 37 39 41 43 47 49 211 3 7 3 29 51 S3 57 59 61 63 67 69 71 73 77 79 81 83. 87 89. 91 93 97 99 29 3 157 3 662 663 3 239 3 11 73 3 13 23 17 3 11 47 7 103 97 11 59 173 113 3 11 43 31 41 3 197 664 127 3 181 3 41 13 665 227 3 11 3 103 '7 11 61 19 101 139 11 i7 666 667 668 3 137 11 3 241 3 101 179 23 17 67 71 3 109 3 47 211 3 151 3 7 10 9 669 149 3 23 3 29 3 167 3 193 11 670 11 23 13 229 17 9 9 671 239 23 672 17 3 109 3 103 3 137 3 11 10 673674 47 3 193 3 675 191 109 3 181 107 251 257 3 23 3 10 676 241 ii 47 239 11 11 71 157 7 53 3 113 3 677 678 679 3, 79 3. 11 7 59 3 113 3 23 29 3. 7 3 179 13 19 41 13 3 3. 11 79, 3 67 13 103 7 3 101 3 3. 11 3 151 11 (157 3 3. 29 97 53 12 190 BRANCKER'S TABLE OF COMPOSITE AND 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 01 03 07 09 47 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 251 59 13 7 241 3 167 3 83 3 il 3 193 17 11 83 23 89 61 23 3 127 3 3 131137 19 3 233 7 223 3 7 3 47 3 31 13 163i 3 J- 9\ 6 173 149 53 107 7 83 13 13 11 73 257 47 7 7 37 107 11 103 3 7 3 11 23 3 223 3 43 19 37 251 23 151 67 43 11 157 11 11 7 257 17 211 3 7 8 127 59 3 17 3 227 101 71 7 223 14 13 151 139 29 23 113 11 107 3 11 167 19 7 10 INCOMPOSITE NUMBERS LESS THAN 100,000. 191 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 S3 57 59 61 63 67 69 71 73 77 79 81 83 87 89 41 91 93 97 99 10 13 17 12 3 163 3 7 7 229 167 3 23 3 181 13 13 7 107 151 97 3 241 3 47 251 3 157 3 11 13 163 3 3 73 3 227 7 197 107 3 13 3 127 101 3 11 3 19 17 223 7 11 19 11 3 263 3 139 173 13 83 31 11 29 3 251 3 97 3 211 3 227 4] 7 13 3 17 3 229 19 17 233 7 3 191 3 11 3 179 3 3 17 3 13 10 9 9 7 9 14 103 109 17 3 263 3 13 149 3 137 3 13 3 7 3 101 29 7 13 157 109 29 19 13 3 181 3 59 3 229 3 3 7 3 257 11 11 83 3 37 13 11 11 23 71 17 11 11 7 47 227 3 79 3 167 3 193 3 17 3 11 7 9 9| 13| 11 9 6 8 12 11 192 BRANCKER'S TABLE OP COMPOSITE AND 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 01 03 07 09 89 3 13 3 "7 3 103 3 163 17 3 "3 7 17 61 19 3 ."3 31 79 3 17 3 '23 "7 3 47 3 11 "3 "3 37 7 11 3 41 3 29 71 3 19 3 23 'is 3 11 3 7 31 3 7 3 11 89 3 7 3 7 3 23 3 67 263 'ii 11 13 17 19 107 23 11 3 37 3 41 "3 257 3 167 "7 13 3 11 3 139 59 3 127 3 7 ioi 3 19 3 17 3 "3 "i7 13 7 3 "3 113 3 11 3 179 7 211 17 3 167 3 157 13 3 19 11 3 "3 7 11 3 7 3 31 223 97 3 7 3 193 3 37 21 23 27 29 3 7 3 17 7 3 11 3 3 31 3 151 "3 23 3 47 11 7 29 3 "3 59 11 3 "3 7 "ig 67 3 "3 233 13 3 103 3 '83 "'7 3 37 3 13 17 3 "3 "7 101 97 3 "3 83 3 17 3 'is 'ii 3 "3 7 29 3 7 3 31 33 37 39 "3 7 3 17 53 13 3 7 3 29 7 3 "3 113 17 107 3 '3 17 13 3 19 3 257 "7 3 173 3 13 "3 7 199 3 "3 11 67 3 "3 • is 11 7 3 "3 23 23 3 151 3 29 7 2ii 3 11 3 19 17 3 47 3 11 17 107 3 ... 41 43 47 49 61 i09 3 19 3 7 13 3 7 3 '73 11 71 3 7 3 13 7 3 "3 17 3 11 3 23 23 3 97 3 11 13 7 3 "3 17 3 193 3 7 "89 11 3 71 ' 3 41 271 3 11 3 13 251 "7 3 "3 47 37 3 29 3 41 7 3 "3 73 51 53 57 59 61 63 67 69 71 73 77 79 3 "3 1? 23 3 59 3 ... 3 53 3 7 '13 37 3 7 3 lis 7 3 31 1 263 11 41 3 "3 11 3 43 <1 13 191 7 HO 3 17 3 "'3 109 1 7 3 "3 73 3 m 7 3 13 3 "3 13 3 19 11 3 7 17 3 17 11 3 19 3 97 "7 "'7 3 127 3 S69 3 933 3 149 3 7 31 3 "3 "3 7 3 'ii 13 '53 3 61 3 7 3 7 3 "3 7 3 7 3 131 3 43 '19 '31 89 3 7 3 3 23 3 19 7 3 13 3 61 3 41 3 11 47 i27 "7 3 239 3 3 13 3 11 "3 13 3 7 19 11 23 3 "3 3 17 3 71 "3 11 3 233 3 "3 7 31 is ■37 17 7 3 "3 29 3 13 7 157 3 23 3 11 3 "3 89 "3 "3 3 '"3 29 7 11 81 83 87 89 3 11 3 19 3 37 3 11 41 "7 3 "3 191 "3 173 3 181 7 29 11 3 13 3 73 3 11 3 31 '23 3 59 3 7 107 3 7 3 ii 163 3 7 3 83 7 3 "3 197 '43 13 3 '3 "3 31 3 89 3 167 3 241 3 7 113 3 37 91 93 97 99 "3 17 3 7 11 23 17 3 13 3 197 11 3 13 3 71 "7 3 229 3 19 157 3 139 3 83 7 '43 3 '3 269 47 3 "3 "19 67 13 3 53 3 7 '■3 7 3 79 93 19 29 3 7 3 67 7 3 "3 59 is 3 109 3 11 19 3 3 23 61 7 9 8 11! 8 8 6 11 9 8 12 10 6 5 10 8 10 11 7 9 7 INCOMPOSITE NUMBERS LESS THAN 100)000. 193 740 741 742 743 744 745 746 747 748;749;750 I 7511752 753 754 755 756 757 758 759 01 03 07 09 11 13 17. 19 21 23 27 29 31 33 37 39 3 '3 13 ii 3239 3 269 3 43 41 11 43 " 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 151 53 149 7 3 23 31 131 3 43 13 191 101 1 "25" 3 19 3 263 73 19 131 3 7 3 127 23 3 211 3 173 ii 19 3 197 3 7 163 3 23 3 19 3 113 3 8 11 131 19 239 7 11 3 3... 137 3 103 3 11 3179 3 107 3 257 .. . 3 3 149 3 241 17 23 7 3 271 3 12 111 6 227 29 3 163 3 223 3 17 3 11 41 37 193 3 73 127 7 11 109 19 11 3 179 43 199 3 11 3 241 7 197 3 61 3 13 11 17 103 10 10 11 11 131 269 83 17 11 191 13 11 89 31 3 23 3 7 3. 181 3 211 7 239 149 3 73 3 107 7 173 53 3 151 3 13 37 3 17. 17 3 23 3 3 "3 19 229 13 3, 29 3. 71 12! 10 12 10 194 BRANCKER'S TABLE OP COMPOSITE AND 760 761 01 03 07 09 11 13 17 19 21 23 27 29 163 269 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 139 13 3 11 3 113 271 7 3 127 3 81 83 87 89 91 93 97 99 8 7 10 762 763 764 3 17 3 167 7 , 3. 127 3 7 79 241 19 7651766 3 59 3 103 11 3 227 113 19 13 767 768 53 271 23 59 769 101 3 7 3 17 3 131 3 7, 9 23 3 167 3 770 771 53 17 3 233 3 13 13 29 251 263 772 773 83 37 3229 7 71 113 13 774 775 776'777|778 17 19 3 17 11179 3. 167 3 211 3 223 19 13 3 193 3 7 199 41 11 7 12! 9 9 13 10 11 11 71 7 11 149 131 13 23 17 59 3 223 3 13 7 277 3 107 779 3 7 3 17 13 17 3 7 3 127 3 13 3 11 67 29 149 3 59 41 3 23 3 137 11 7 3 53 3 103 3 29 h 167 3 23 3 INCOMPOSITE NUMBERS LESS THAN 100,000. 195 780 T81 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797798 799 01 03 07 09 11 13 17 19 181 13 'ei 21 23 27 29 3. 11 3 7 31 33 37 39 41 43 47 49 51 53 57 59- 61 63 67 69 251 3 11 3 71 73 77 79 81 83 87 89 91 93 97 99 11 19 137 71 17 3 13 29 7 3 233 17 19 11 107 41 31 157 47 47 7 101. 163 3 113 3 11 7 11 29 131 3 13 103 3. 43 277 3 11 3 7 211 31 13 3 127 11 3 223 3 251 3 179 3 11 3 7 13 53 3 269 3 131 43 7i 17 79 13 227 29 11 3 239 3 137 173 a 37 3 107 3 7 257 10 139 7 19 83 7 5 3 103 3 11 113 37 7 3 227 3 13 17 11 3 271 3 11 43 107 3 43 3 23 7 131 11 3 281 3 67 11 3 103 13 11 7 181 3 229 3 13 3 251 3 163 3 7 3 3 37 13 3 179 3 7 173 47 19 241 3 19 41 3 157 3 7 229 3 257 3 3 31 17 3 37 3 7 13 3 211 11 12 7 7 14 109 11 3 41 167 13 11 196 BRANCKER'S TABLE OF COMPOSITE AND 800 801 802'803;804 805 806 807 808 809 810 01 3 03 3 163 3 11 3 1391131 3 17 223 7 227 3 127 3 19 83 11 3 181 7 3 283 17 37 191 97 29 137 37 257 11 43 3 239 3 7 179 3 11 3 19 3 211 3 263 13 19 13 3 131 3 229 11 109 .. 61 13 59 79 7 83 19 11 17 233 3 7 3 11 811 812i813 814|815 816 817 818 19 7, 3. 61 3 13 17 13 3 241 3 7 73 3 19 11 173 23 3 41 43 3 11. .1 3 3 3 103 3 7 3 17 277 83 31 193 11 23 17 31 233 7 13 67 3 7 7 3 11 3 149 3 29 101 819 3 179 3 7 71 17 3 47 11 41 257 199 3 23 3 17 127 3 19 227 13 11 29 223 3 11 19 101 13 11 3 17 3 23 109 71 151 3 7 3 67 i9 7 3 41 11 3 3 19. 3 3. 41 53 3 89 263 157 7 73 3 11 3 163 7 3 167 3 10 111 8 6 13 10 8 14 8 10 10 10 7 10 INCOMPOSITE NUMBERS LESS THAN 100,000. 197 820 821 822 823 824 325 526 327 528 329 830 831 332 833 834 835 336 837 838 839 01 03 07 09 43 3 7 3 47 7 3 13 3 19 3 23 17 3 "3 '17 7 3 191 3 11 31 3 17 3 7 3 "3 41 3 19 "7 3 11 3 227 "3 "'3 11 7 113 37 3 13 3 / "3 13 3 47 181 43 11 3 "3 - 7 11 17 3 3 53 11 13 17 19 3 "3 7 157 3 7 3 229 19 3 7 3 263 7 3 73 3 11 109 19 179 3 "3 107 3 181 3 3 17 3 11 3 7 17 "43 3 13 3 "3 13 3 239 11 "7 3 23 3 47 11 3 97 7 3 "3 79 "3 31 3 7 11 3 383 3 ... 21 23 27 29 "3 11 3 13 41 17 3 '"3 7 191 3 7 3 'ii 139 31 3 7 3 7 3 53 3 3 13 3 113 101 3 13 3 61 "7 79 3 101 3 97 "3 "3 7 97 103 23 3 "3 19 17 3 101 3 'ii 241 7 3 29 3 101 109 3 17 3 "7 23 17 31 33 37 39 ".7 3 23 3 "3 "3 17 281 137 3 13 3 7 "3 7 3 19 'i7 23 3 7 3 17 7 3 "3 127 239 197 3 43 3 11 59 3 "3 "7 13 3 167 3 "'3 7 103 3 11 3 31 3 "3 11 3 13 7 3 3 139 41 43 47 49 3 13 3 11 3 67 3 19 3 29 3 59 197 23 3 11 3 7 97 3 7 3 11 37 '13 3 7 3 109 7 3 "3 71 29 17 11 3 "3 17 "3 11 3 181 "7 3 19 3 29 "3 233 3 7 11 83 89 3 '3 191 11 3 127 3 3 13 3 7 11 233 51 S3 57 59 "3 31 3 113 '29 7 3 83 3 43 "3 11 3 41 7 'is 3 31 3 "3 83 11 3 29 3 7 11 3 7 3 53 23 13 3 7 3 137 7 3 "3 17 19 'si 3 17 3 13 3 "3 23 "7 269 3 61 3 13 71 3 "3 7 37 59 113 3 ... 61 63 67 69 71 73 77 79 7 137 is 3 "3 211 3 "3 127 "3 37 3 "3 3 '23 31 7 3 '■3 11 3 "3 "3 131 7 13 19 3 "3 37 41 3 173 23 3 13 3 7 139 53 3 7 3 11 7 3 19 3 'ii 193 3 "3 SI "3 211 3 17 13 7 3 11 3 163 3 7 7 29 13 "'3 67 3 '71 11 7 3 47 3 29 13 3 23 3 79 7 179 67 3 11 3 13 "3 "3 11 31 223 3 "3 7 263 3 7 3 'is 'ii 3 7 3 7 3 11 3 19 3 131 3 79 3 m 3 37 / 81 83 87 89 79 3 23 3 11 "7 ,3 107 3 19 13 3 7 3 269 3 13 89 3 11 3 '19 "7 3 "3 • • • 3 31 3 251 7 19 3 193 3 41 11 3 37 3 199 'ei 3 31 3 7 19 3 7 3 13 67 53 3 7 3 23 7 3 149 3 137 '47 3 11 91 93 97 99 103 11 53 19 3 "3 13 11 3 17 3 47 3 "'3 151 3 7 13 41 3 "3 "3 19 3 37 149 "'7 3 "3 23 23 3 271 3 13 7 31 3 89 3 29 3 "3 i79 "i'l / 3 127 3 7 "3 7 3 '43 11 53 3 7 3 19 7 17 3 11 9 10 9 9; 9| 8 11 7 6 10 4 13 7 I2I 7 8l 8 7 8 193 BRANCKER'S TABLE OF COMPOSITE AND 840 841 842'843 844 845 846 847'848 849 850 851 852 853 854 855 856 857 858 859 01 167 37 3 7 ... 3 11 T7T 3 59 7 3 ... 197 3 13 3 2.39 17 03 3 31 7 3 ii , , , 3 7i 137 3 167 ,, . 3 41 3 "i 3 • • • 07 7 151 3 • • . 3 19 7 3 197 13 3 139 23 3 37 * • > "3 53 271 09 3 241 107 3 13 3 23 3 3 7 223 3 59 13 3 11 3 59 3 7 211 3 19 3 13 7 3 233 3 11 3 13 29 19 "3 7 3 191 "3 151 3 3 11 "3 7 53 17 3 7 3 223 13 3 89 '7 3 47 "ii "3 229 '.. . 3 3 19 "is 7 3 ... 29 3 37 3 11 3 31 13 3 "i "3 ... 151 21 23 3 73 3 37 3 7 7 3 11 271 ^ 3 163 3 41 3 7 13 3 "3 23 11 3 19 11 3 "3 ■ • ■ "3 ■ • • "3 '23 27 3 . • . 11 3 7 181 3 193 3 • ■ • 7 "3 11 • t • "3 • > . 59 3 29 29 11 3 ... 7 3 137 ... 3 41 13 3 11 3 ... 31 3 7 ... 3 31 33 17 3 7 3 131 13 3 '23 3 3 7 7 3 23 13 3 29 3 ... 3 37 "3 7 19 3 "3 'ii "3 !! 37 19 • > • 3 11 "3 7 3 157 • • . "3 • ■ . "i 3 23 29 "3 . • . '19 39 3 11 ... 3 ii 7 3 101 43 3 277 19 3 61 ... 3 ... 83 3 7 41 31 3 61 19 3 17 53 3 37 29 3 7 13 3 43 113 3 179 7 3 43 229 • ■ • 3 7 • • • 3 13 83 3 173 7 3 • ■ ■ 31 3 131 • • • 3 t • • .11 47 3 3 59 47 3 7 3 .. . 3 7 3 19 3 49 i 13 "3 ... 3 7 3 'ii '3 i63 11 "3 41 3 a'93 61 51 3 19 173 3 79 3 13 3 17 11 3 7 3 97 3 23 53 . . • 3 13 67 3 "7 "3 53 11 3 17 7 3 • > • 13 3 '29 3 57 3 23 109 3 . . . 11 "'3 131 3 7 31 3 17 97 3 11 7 3 43 59 61 3 7 11 3 3 7 3 3 11 67 3 191 23 3 7 3 29 13 3 31 3 3 11 3 7 3 19 67 63 "3 3 103 3 "7 113 "3 'ii 13 3 7 3 ii 139 3 31 67 'i7 "3 239 3 11 29 3 257 3 7 'ig 3 41 3 17 7 69 "3 73 17 3 ' 7 19 3 103 ... "3 97 7 3 ... ... 3 ... 199 3 13 71 13 3 11 7 3 23 227 3 31 3 53 71 3 127 3 7 43 3 73 11 41 3 139 17 3 13 '3 7 241 3 269 59 3 '83 7 3 79 149 77 7 3 71 3 83 'ii 3 13 ■ • > 3 19 53 3 7 • ■ • 3 31 11 3 79 83 ... 3 'ig 23 3 7 17 3 ... 149 3 107 7 3 13 11 3 157 127 81 3 271 3 7 3 149 17 3 103 3 11 3 47 3 7 83 47 "3 89! 13 "3 41 19 3 29 17 "3 7 11 "3 73 23 3 i09 7 3 87 3 29 7 3 13 251 3 11 3 . • . 17 3 103 > • • 3 7 13 3 11 89 ... 3 31 ... 3 11 3 7 37 3 13 17 3 53 7 3 11 ... 3 91 7 3 11 3 7 3 3 19 17 3 11 3 13 93 3 '59 11 "3 19 29 "3 23 "3 3 7 17 3 'ei 3113 97 13 269 3 371 ^ 3 . > . "19 3 11 '43 "3 13 3 17 "3 7i 23 99 3 3 31 3 11 73 3 7 ... "3 23 193 "3 43 7 3... _I 8 10 8 9 12 8 8 9 7 8| 10 8 9 _8 9 8 11 7 9 6 INCOMPOSITE NUMBERS LESS THAN 100,000. 199 )861 862 863 864 865 866 867 869 870 871 872 873 874 875 876 877 878 879 01 03 07 09 11. 13 17. 19 11 21 23 27 29 31 33 37 39 3. 227 3. 97 151 3 23 3 37 13 m 173 131 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 83 87 89 91 93 97 99 11 277 7 7 28i 59 si' 19 211 3 7 3 241 11 3 11 3 257 277 3 31 3 3 197 3 13 73 7 193 19 127 223 13 3 101 3 43 3 233 17 227 3 7 3 7 173 29 3 151 3 3 263 3 7 283 3 17 3 131 13 7 229 3 29 3 7 31 J 13 67 17 3 251 3 71 199 3 7 3 3 157 11 149 47 23 11 12 11 8 10 71 9 13 17 11 7 10 10 281 3 1 3 13 3 11 3 251 7 229 139 3 277 11 3 17 3 79 3 239 3 3 127 3 3 137 3 13 3 107 3 13 3 11 3 23 3 47 3 31 37 59 '7 103 3 281 3 7 19 3 13 29... 3 11 11 3 11 3 13 23 179 97 3 7 3 11 3 19 ' 3 7 3 13' 12 200 BRANCKER'S TABLE OF COMPOSITE AND 880 881882,883 884 I 885 887888 890 891 892 893 894 895 896897 898 899 01. 03 07 09 17 11 13 17 19 3 283 3 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 193... 3 227 7I233 3 13 3 17 47 19 3 47 7 71 73 77 79 81 83 87 89 91 93 97 99 89 3 7 3 211 29 61 41 251 11 223 3 29 3 263 61 151 137 3 59 23 3 3241 17 3 53 3 149 3 31 11 61| 97 3 19 7157 3! 37 111 3109 3 103 3 7 S3 101 283 19 3 131 3 211 3 7 3 19 73 3 7 3 269 17 3271 .. 3 3109 13 3 233 13 97 239 59 179 13 7 3 223 3 37 47 13 3 103 3 281 3 101 3 23 3 229 163 3 13 3 23 7 11 257 3 101 . 3 149 3 7 3 199 11 19 193 3 151 3 149 17 3 157 3 37 283 13 73 7 137 3 7 3 61 79 191 29 71 13 3 43 3 109 43 29 3 101 3 139 3 3 7257 3 19 3 11 3 47 3 13 19 7 11 3 139 3 53 3 11 3 293 23 7 23 73. 13 3 11 3 7 3. 17 11 17 '29 13... 3 3 241 31 . 3 4 7, 7 5 9 8 13 11 8 101 9 8 12 11 6 10 9 INCOMPOSITE NUMBERS LESS THAN 100,000. 201 900901902 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 11 13. 3 97 3 227 3, "3. 197 179 41 43 47 49 51 53- 57 59- 193 173 23 7 3 109 3 61 63 67 69 113 3 37 71 73 77 79 81 83 87 89. 91 93 97 99 903 73 3 7 3 13 37 181 103 3 13 3 29 31 11 7 3137 17 904 109 3 23 3 11 905 3 7 3 29 3 149 3 151 131 906 7 3 11 3 907 908 13 61 19 31 257 3 7 3 7 3 197 3 3 233 3 11. 3 3 ri03 3 13. 23 83 137 7 3 269 3 151 3 47 3 13 11 17 139 7 3 173 53, .239 7 157 23 29 909 23 19 3 7 3. 11 3 17 3 3 '3 229 23 13 Ml_ "26" 910 17 11 7 7 3 227 3 211 199 ios 89 29 911912913 179 3 13 3 293 3 181 3 7 61 n 3 223 3 3 97, 163 "7 29 19 3 8 10 29 3 271 3 11 149 241 23 11 13 263 3 11 3 107 7 97 37 19 67 914 3 13 3 17 3 7 3 167 103 211 11 3 '3 915 37 3 13 3 3 113 3 109 916 139 47 101 7 11 3 239 3 17 917 3 293 7 3 151 3 67 3. '3 7 17 3 277 918 "3 '3 919 29 7 73 3 229 (131 3 3. 199 11 3 7 3, 163 3 107 3 17 3 11 3 7 149 89 3. 31 3. 97 13 "79 139 17 263 19 59 "7. 7 47 107 41 59 3 7 3 67 11 197 12 8 10 9 10 202 BRANCKER'S TABLE OP COMPOSITE AND 920 921 922 923 924,925926 927 928 929 930 931 932933 934 93S 936937 93d 939 01 03. 07 09. 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 71 73 77 79 81 iB3 87 89 7 251 137 3 241 19 13 19 7 17 127 61 13 257 7 19 3. 3. 7] 91 93 97 99 7i.. 3' . 13.. 3 11 233 79 23 211 3; 7 53 7 -.i' 3] 17; 29 3 7 9j 10 13| 9 13 13 17 263 3 227 3 13 3 19 41 3 11 • 3 3 293 3 29 3 239 3 109 3 12 10 11 11 29 3 109 3 179 31 269 17 277 13 233 223 41 113 3 7 3 19 3 13 179 17 3 251 3 83 19 17 3 211 3 11 229 3 7 3 47 283 113 23 3 13 3 241 3, 23 3 7 17. loi 3 103 3. 107 11, 3 13 3. 127 17 47 3 7 3. 37 269 223. 1193 3 3... 13 3 10 10 10 8 5 8 12 INCOMPOSITE NUMBERS LESS THAN 100,000. 203 01 03 07 09 940 23 7 941 3 139 3 11 13 17 19 21 23 27 29 31 33 37 39 3. 41 3. 149 167 3 17 3 271 11 41 43 47 49 3 157 3 51 53 57 59 61 63 67 69 11 m 19 71 73 77 79 81 83 87 89 91 93 97 99 163. 3. 942 "3 "3 3 37 3 257 3 107 3 943 181 11 8 11 944 3 67 3 7 19 3 263 3 7107 945 11 3 7 3 29 '47 31 17 '7 271 11 946947 13 37, 11 43 173 7 101 17 3 61 3 211 3 103 3 3 193 3 97 3. 17 3. 13 23 281' 11 948949 7 3 113 3 43 107 59 ,139 13 3 269 3 3 239 3 13 950 3 , 3, 167 11 101 3 17 3 951 "3 '3 227 11 73 3 7 3 251 952 31 "7 19 131 47 151 3 7 11 233 157 12 953 3 13 3 191 283 3 127 3 954 73 955 7 43 149 7 307 13 227 956 7 3 101 3 3 271 3 7 29 3 241 3 163 7 103 11 957 61 47 7 239 958 257 17 37 959 3 29 3 11 13 3 23 3 197 229 11 204 BRANCKER'S TABLE OF COMPOSITE AND 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 01 03 07 09 11 13 17 19 67 3 223 3 277 149 17 229 21 23 27 29 31 33 37 39 41 43 47 49 3 131 3 109 7 251. i27 13 3 211 3 11 "7 139 51 S3. 57 59. 61 63 67 69 71 73 77 79 23 191 29 17. 3173 3 29 81 83 87 89 91 93 97 99 307 3 263 7 83 3 7 3 131 223 269 3 13 3 59 241 127 7 277 7 11 3 41 47 31. ^1- 3 13 37 3 103 3 7 11 3 13 3 157 11 23 3 293 3 3 193 41 67 191 3 23 3 307 13 131 19 11 3 257 3 13 3 281 3 3 41 3 199 47 13 19 7 17 311 11 3 271 3 311 3 7 41 3 233 3 17 89 163 251 3 179 3 181 3 19 3 227 3 7. 3. 13. 3. 3 41 11 3 107 3 10' 6! 12 6! 12 7 17 173 10 239 67 11 29. 3. 59 3 7 7 3 23 3 163 97. 13 277 7. 13 3 11 211 151 7 3, 11 53 3 223 3 11 29 7 43 11 INCOMPOSITE NUMBERS LESS THAN 100,000. 205 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 01 03 07 09 11 13 17 19 21 23 27 29 31 33 37 39 41 43 47 49 51 53 57 59 61 63 67 69 283 7 17 3 7 3 281 71 73 77 79 101 3 7 3 81 83 87 89 91 93 97 99 3 197 3 37 13 137 23 3 151 3 7 257 3 173 3 37 2ii 7 19 17 3 269 3 29 7 97 3 11 891 13 3 19 139 3 67 3 3 233 3 263 149 3 11 7 241 23 227 13 7 3 311 3 293 19 11 7 13 3 283 13 109 43 11 83 167 3 97 3 53 11 3 61 173 3 223 3 7 3 229 11 13 313 67 13 199 3 13 3 13 3 229 3 197 41 41 281 7 19 8 12 11 8 11 3 277 3 7 191 3 11 3 23 11 3 271 3 3 37. 3 43 3 37 251 11 113 29 227 3 7l 79 7 17 3 17 11 7 263 151 3 19 139 17 89 127 23 31 67 19 131 73 3 7 9 23 59 23. 3 7 3 19 257 17 11 3 13 3 3 191 3 283 8 10 10 3 19 3 1 20G TABLE OF POWERS. TABLE OF POWERS. 4» 2 2» 2' 2< 2" 2« 2' 28 2« 210 45 2" 212 46 213 211 47 215 216 48 217 218 49 218 220 410 221 228 411 223 224 412 8 16 43 82 i* 16« 8 16 32 64 128 256 8' 512 1024 2048 8* 16= 4096 8192 16384 8= 32768 65536 131072 262144 524288 1048576 2097152 4194304 8388608 88 16» 16777216 92 9< le' 8« 16» 8' 3 3^' 3' 3* 35 3« 3' 38 3» 310 95 3" 31s ge 313 3" 9' 316 318 9« 317 3'8 9» 319 320 910 321 322 911 323 324 91s 27 81 93 27" 8V 27' 27* 81» 27 81 243 729 2187 • 6561 19683 59049 177147 531441 1594323 4782969 14348907 43046721 129140163 387420489 1162261467 3486784401 10460353203 31381059609 94143178827 27» 81* 282429536481 27» 81* 27« 8F 27' 5 5* 5» 5* 55 5« 5' 58 59 510 5" 5" 5 25 125 625 3125 15625 78125 390625 1953125 9765625 48828125 25» 244140625 25 25« 25' 25' 25» 7 7« 7' 74 76 7» V 7' 7» 710 496 49 492 49' 49* 7 49 343 2401 16.8P7 117649 823543 5764801 40353607 282475249 1977326743 7" 4^» 13841287201 11 11 IP 121 11' 1331 11* 14641 IP 161051 11" 1771561 ir 19487171 11' 214358881 IP 2357947691 ll'" 25937424601 11" 9853116706U 11'^ 3138428376721 13 . l3^ 13'. 13*. 13 169 2197 28561 135 371293 13° 4826809 17 17 17' 289 17' 4913 17* 83521 17' 1419857 17« 241375G9 TABLE OP POWERS. 207 TABLE OF POWERS. A Table of all the composite numbers ending in 1, 3, 7 or 9, be- tween 20,000 and 1,000,000, which are perfect powers of any prime number. The prime number roots annexed with their indices ex- hibit all the prime factors of their powers. The table of the prime factors of odd numbers shows the roots of all these numbers below 20,000. This table, in regard to the numbers comprised in it, is de- signed to dispense with the use of the trial Rule given under Sect. IV, page 34, for finding the prime factors of certain numbers above the limits of the tables. POWERS. ROOTS. POWERS. ROOTS. POWERS. ROOTS. POWERS. ROOTS. 22201 149= 357911 713 24389 29' 351649 593= 22801 151= 358801 599= 24649 157= 368449 607= 28561 13" 361201 601= 26569 163= 375769 613= 29791 3P 383161 619= 27889 167= 380689 617= 32041 179= 398161 631= 29929 173= 413449 643= 32761 1812 410881 641= 37249 193= 418609 647= 36481 191= 434281 659= 38809 197= 426409 653= 39601 199= 436921 661= 49729 223» 452929 673= 44521 211= 477481 691= 51529 227= 458329 677= 52441 229= 491401 701= 54289 233= 493039 79' 57121 239= 502681 709= 66049 257= 528529 727= 58081 241= 516961 719= 69169 263= 537289 733= 63001 251= 546121 739= 76729 277= 552049 743= 68921 413 564001 751= 80089 283= 573049 757= 72361 269= 579121 761= 85849 293= 597529 773= 73441 271= 591361 769= 94249 307= 619369 787= 78961 281= ' 654481 809= 97969 313= 635209 797= 83521 17" 657721 811= 100489 317= 677329 823= 96721 311= 674041 821= 113569 337= 683929 827= 109561 331= 687241 829= 120409 347= 704969 89' 121801 349= 703921 839= 124609 353= ■ 727609 853= 128881 359= 707281 29" 139129 373= 734449 857= 130321 19" 737881 859= 146689 383= 744769 863= 136161 369= 776161 881= 157609 397= 769129 877= 143641 379= 829921 911= 187489 433= 779689 883= 151321 389= 844561 919= 196249 443= 786769 887= 160801 401= 863041 929= 205379 59' 822649 907= 167281 409= 885481 941= 253009 503= 877969 937= 175561 419= 923521 31" 273529 523= 896809 947= 177241 421= 942841 971= 299209 547= 908209 953= 185761 431= 982081 991= 310249 557= 935089. 967= 192721 439= 50653 103823 300763 371293 373 473 67' 13^ 316969 563= 954529 977= 201601 449= 332929 577= 966289 983= 226981 259081 271441 613 509= 521= 344569 587= 999009 997' 279841 23" 79507 433 292681 541= 148887 53' 323761 569= 389017 73» 326041 571= 571787 83' 208 SOME RELATIONS OF NUMBERS. OP THE RELATIONS BETWEEN THE FACTORS, THE ALIQUOT PARTS, AND THE DIVISORS OF NUMBERS. The factors, prime or composite of any number, do not include unity or the number itself. The aliquot parts of a number are all its factors prime and composite and unity or 1. The divisors of a number are all its aliqnot parts and the number itself If the index of each of the prime factors of any given number be aug- mented by 1, then the product of all the indices thus augmented will be the number of all the divisors of the given number; and this, less 1, will be the number of its aliquot parts; and this, less 1, will be the number of all its factors prime and composite. OF NUMBERS PERFECT, ABUNDANT, DEFICIENT AND AMICABLE. These numbers are comparatively very few. They are defined by Eu- clid and other authors with reference to the magnitude of the sums of their respective aliquot parts in comparison with that of the numbers themselves. A perfect number is one which is equal to the sum of all its aliquot parts. An abundant number is such that the sum of its aliquot parts is greater than itself A deficient number is such that the sum of its aliquot parts is less than itself Two numbers are called amicable, if the sum of the aliquot parts of each one is equal to the other number. For methods of obtaining numbers of each of these classes, see Euclid's Elements, Bonnycastle's Arithmetic, and Taylor's Theory of Numbers. OF NUMBERS EVENLY ODD, EVENLY EVEN, AND UNEVENLY EVEN. A number is evenly odd if, upon its being divided by 2, the quotient be an odd number. A number is evenly even, if it and each successive quotient be divisible by 2, until the last quotient is 1. A number is unevenly even, if upon its being divided by 4 or some higher power of 2, the quotient be an odd number greater than 1. These three classes comprise all even numbers. The evenly odd num- bers, inclusive of 2, comprise one half of the even numbers of any even number of terms of the natural series. They are distinguished by their terminations, as shewn in Table I, on page 29. Moreover in the table oif the factors .of even numbers the first factor of every evenly odd number within its limits, is 2. The evenly even numbers comprise only the powers of 2. All even numbers not evenly odd or evenly even are unevenly even. Table II on page 29, comprises both the evenly even and the unevenly even numbers. In the table of the factors of even numbers, each number-" that is unevenly even has 2" or some higher power of 2 for its first factor, and an odd number or odd numbers for its other factor or factors. Equal ratios cannot be formed of prime numbers. APPENDIX. After the several parts of this book were composed and stereotyped, the author from his own "'observation upon the tables, and from several mathematical works which he consulted, obtained a knowledge of some things relative to prime numbers which he deems of sufficient interest to be inserted here by way of Appendix. A BRIEF NOTICE OF SOME AUTHORS AND TABLES. Prime numbers engaged the attention of, at least, two distinguished mathematicians among the ancients; and they have taxed the ingenuity of many among the moderns. Euclid, who flourished three hundred yeiars before the Christian era, well understood many of their properties; as is evident from what is contained respecting them in the seventh, eighth, and ninth book of his Elements. Eratosthenes, who succeeded him by about a century, is said to have been the first who invented a method, which he called his sieve, for sepa- rating them from other numbers. Thomas Taylor, in his Theoretic Arithmetic, London edition of 1816, has furnished, in a tabular form, a representation of the sieve, a copy of which is here inserted. (See page 211.) This sieve is unnecessarily burdened with numbers ending in 5. The principle of this sieve, or that of the table on page 17, may be applied separately to different classes of numbers ; as for instance to all those ending in either one of the digits 1, 3, 7, 9; or to classes more limited as those ending in 101, 201, 301; or 103, 203, 303, 1001, 2001, 3001, &c., it being observed that the numbers assumed must have a com- mon difference of 10, 100, or 1000, &c., so as to correspond to the deci- mal scale. Many modem mathematicians, it is said, have in vain exercised their skill in the endeavor to invent a rule for the discovery of prime numbers by a direct process. Upon this point, Peter Barlow in his book enti- tled New Mathematical Tables, published in London, in 1814, says: "A method of finding a prime number beyond a certain limit, by a d. ect process, is considered as one of the most difficult problems in the theory of numbers; which, like the quadrature of the circle, the trisection of an angle, and the duplication of the cube, has engaged the attention of many of the ablest mathematicians of modern times." 27 209 210 APPENDIX. Legendhe, in the second edition of his work entitled, '^Essai sur la Tkeorie Des JVombres,^'' on page 11 of the Introduction, says: "En gene- ral il n'existe aucune formule algebrique propre d n'exprimer que des nombres premiers." There does not exist any general algebraic formula •proper to express prime numbers only. Nevertheless these numbers have been discovered to a very great extent by other means, so that such a process or formula would now be of little practical importance. Various tables have, within the last two centuries, been made and published in Europe; some of which consist of prime numbers only, others of these numbers and also of some or all of the simple divisors or prime factors of the composite numbers, within their limits. The oldest one of the tables, of which the writer has obtained informa- tion, is that published as an Appendix to the Algebra of Rhonius, at Zu- rich in Switzerland, in 1659. This was the foundation, or first part, of Brancker's long table, published in England, with a translation of the Al- gebra, in 1668. A large part of this table, revised, has been copied into this book. For a more particular account of it see page 163. Near the end of the eighteenth century, George Vega, a distinguished mathematician of Germany, published a work in the Latin and German language, containing two tables, one of prime numbers and factors and the other of prime numbers only. The precise date of the first edition of this work is not known. But on the fourth day of October, 1852, in response to a letter of inquiry for this work and others mentioned by Legendre, the writer received a copy of the second edition of it, with the interesting information, that this copy, which it appears belonged at some time to Glasgow College, was comprised in the mathematical library of Mr. Jacobi ; which was bought entire for the library of Har- vard College, by Mr. Bond the younger, of the Astronomical Observatory at Cambridge, on his travel to the North of Europe to observe the great eclipse of the sun in 1851. Fortunately this copy of Vega, which is probably the only one that has reached this country, came to hand just in time to be properly noticed here, and to serve as a further test of the accuracy of the tables of this book. The title of the work in Latin is, "Tabula Logarithmico- Trigonometric*, cum diversis aliis in Matheseos usum constructis Ta- BULis ET FoBMULis." It has also a title of the same import in German. The second edition was piiblished at Leipsic in 1797. It is styled — "Editio secunda, emendata penitusque reformata." The tables above alluded to are contained in the second volume. The first is entitled, " Tabula omnium divisorum simplicium numerorum per 2, 3, 6, non divisibilium, ab 1 usque ad 102,000. Factores litteiis a, b, c, d designati indicant numeros primos, 11, 13, 17, 19." 211 Primary and In- composite Numbers. 11 IF 17 l9~ 23 29 IT 37 41 47 53 59 IT 67 71 Odd Numbers. The series of Odd Numbers which are measured by 3. 3 5 7 9 9 11 13 15 15 17 19 21 21 . 23 25 . 27 27 29 31 33 33 35 37 39 39 41 43 45 45 47 49 51 51 53 55 57 57 59 61 63 63 65 67 69- 69 71 73 75 75 77 & t § i- s? Bl' 3 c The series of Odd Numbers which are measured by 5. 15 IB S ?, m 25 s 35 a. CO o ^ 45 's- s- 55 o FT j-5 65 pi m O O CD S 75 I n 3 ■cq' 3 i I The series of Odd Numbers which are measured by 7. a- 21 en a. f o CO B' CD CD 2 35 s p. D- "«« en o &■ 49 m s- cn Pi 63 o p. O o 3 77 K 212 APPENDIX. The second is entitled, "Continuatio numerorum frimorum, as 102,000, USQUE AD 400,000." These tables together furnish a list of prime numbers from 1 to 400,031. The first, which may be called a table of factors, contains no even nimiber nor any odd number divisible by 3 or 5. It designates all the prime numbers within its limits, by blank dotted lines; and it contains, or de- signates by figures or by letters and figures, all the factors of every com- posite number ending in 1, 3, 7 or 9, which is divisible by 7 or any greater prime number. No factor has any index or exponent affixed to it, the power of a factor being expressed by the factor itself repeated. The second table consists of prime numbers only, from 102,001 to 400,031, inclusive. The description given of these tables by Legendre, on page 5 of the Introduction to his Essai, above mentioned, seems not to correspond in all respects to the true character of them, as they appear in the second edition. Speaking of prime numbers, he says : "Dans un Livre intitule, Georgii Vega TabulcB LogarUhnico-Trigo- nmnetrieeB, Lipsia 1797, on en trouve une qui s' etend jusqu' a 400,000, et qui a, de plus, 1' avantage d' indiquer pour chaque nombre compose le petit nombre premier qui en est diviseur." This description, in reference to the tables in the second edition of Vega's work, the date of which is correctly given by Legendre, is errone- ous in three points. It supposes but one table when there are two. It represents the table as one of factors, as well as of prime numbers, through- out the whole range from 1 to 400,000; whereas there are no factors of any number greater than 101,997; an3 that it indicates only the least divisor or factor of each composite number, when in truth, it furnishes in the manner above mentioned, all the factors of not less magnitude than the number 7, of each of its composites. The table of factors is ingen- iously arranged by the limits of centuries, in three different successive forms on every two pages, so as to include all the prime numbers and at the same time to exclude all numbers divisible by 3 or 5. Legendre informs us that Ladislaus Chernac, profegsor of philosophy at Deventer, published in 1811, under the title, Cribrum ArUhmeticum, a table of prime numbers and divisors of other numbers from 1 to 1,000,000 and upwards: and that Jean Charles Burckhardt published at Paris in 1814, tables of the divisors of all numbers from 1,020,000 to 2,028,000, and that he intended to make a similar table for the third million. This it appears he afterwards did; for we are fiirnished vidth the following title of a work by the same author, viz. "Table des diviseurs pour tons les nombres du 1" 2* et 3« million, avec les nombres premier qui s'y trou- vent, Paris, 1817, gr. in 4to.'' 1 9693 3 8392 3 8014 4 7862 5 7677 6 , 75S5 7 ' 7442 8 7402 9 7331 10 7225 APPENDIX.. 213 OF THE COMPARATIVE NUMBER OP PRIME NUMBERS IN EftUAL SUCCESSIVE INTERVALS. The table on page 20 shews the number of primes (2 and 5 excepted) in each successive cJiiliad and myriad from 1 to 100,000, as well as the whole number within these limits. From Vega's table and the numbers given by Legendre from Chernac's table, we here exhibit the number of primes for each interval of 100,000 from.l to a million. „^ Intervals.. Number of Primes. Differences. 1201 378 isa 185 122 113 40 - 71 106 78493 2368 The ratio of primes in the first interval is 9.593 per cent., while in the last it is only 7.225 = 7i nearly. The average on the whole range is 7.8493 per cent. The rate of decrease is very unequal in the equal inter- vals, and it progresses upon the whole in a rapidly decreasing velocity. The cause of the decrease of prime numbers, as well as the variable rate of it in successive intervals, must be the same as that of the corres- ponding or reciprocal increase of the composites within those intervals. This cause may be seen in the comparative numbers and magnitudes of the prime numbers that are introduced into a table like Brancker's, as the least factors of the composite numbers within any given intervals. The number of prime numbers thus introduced for each interval can easily be found. The highest prime number in Brancker's table is 313, being the next integral square root below \/100,000. The highest that would be intro- duced into a similar table for the next equal range or interval of a hundred thousand would be the integral square root next below ,v/200,000. Pro- gressing in like manner to the ,y/l,000,000, we find that the highest prime which is the integral square root next below this root, is 997. And by the table of primes, it will be found that the whole number from 3 to 997 in- clusive, exclusive of 5, is only 166. These are distributed to the ten in- tervals of 100,000 each, from 1 to a million, in the following proportions Intervals. Number of Primes. Sums. 1 63 2 21 84 3 15 99 4 14 113 5 11 124 6 11 135 7 8 143 g 9 153 9 7 169 10 7 166 The increase of the composites in any interval after the first, must be exactly equal to the decrease of the primes in the same interval. The in- 214 A.PPENDIX. crease of the composites however is not in exact proportion to the number of new primes that are introduced into it, for at least two reasons. First, because they are not all brought in at the beginning of the interval, but successively throughout its whole extent. Secondly, because, inasmuch as the comparative numbers of multiples of primes of different magnitudes, within the same limits, are in the inverse ratio of their magnitudes, the number of multiples of any new prime will be less than those of any of its predecessors in the same interval. The successive odd multiples of 997, for instance, differ from each other by twice this number or 1994. In a table containing all the factors of all the odd numbers, every one of its odd multiples would be introduced, successively; but in a table of the least factors of the odd numbers ending in 1, 3, 7 or 9 only, it would occur much less frequently, as it could not appear as the least factor of any number which has any less factor. It appears, by the first of the foregoing tables, that the average rate of decrease in the primes, from the end of the first to the end of the tenth interval, is a little more than twenty-six hundredths of one per cent. ; while from the end of the seventh to the end of the tenth it is only a little more than seven hundredths of one per cent.; so that the numbers of primes for equal proximate intervals of the ascending scale, approach more and more nearly to equality, ad infinitum. THE SERIES OR NUMBER OF PRIME NUMBERS IS INFINITE. The following plausible but inconclusive demonstration is copied from the Penny Cycloposdia, Art. Prime Number. "There cannot be an end of prime numbers; for if so, let p be the last prime number, and let N be the product of all the prime numbers 2, 3, 5, ji. Now every number is either prime or divisible by a prime. But N -|- 1 is not divisible by 2, 3, 5, &c. or p; since it leaves a remainder, 1, in every such division. It is therefore prime, or there is a prime number, N -(- 1? greater than the greatest prime number p^ which is absurd." Upon reading this demonstration it was discovered that N -|- 1 will not always be a prime number; for 2-3-5-7- 11-13 + 1 =30031=59-509. Afterwards the same demonstration was found in the second edition of Legendre's Essai, on pages 11 and 12 of the Introduction, accompanied by a note (1,) in which it is admitted that the demonstration does not hold good in regard to the above mentioned number 30031. Euclid demonstrated the following proposition : TMre are more prime numbers than any proposed number of prime numbers. The demonstration, which is copied from the translation of the Elements made by James Williamson, M. A. Fellow of Hertford College, 4to edition of 1781, is as follows: "Let A, B, C, be the proposed prime numbers, 1 say, there are more prime numbers than A, B, C. For let the least niun- APPENDIX 2 1,', ber measured by A, B, C, be taken, and let it be DE; and let unity, DF, be added to DE; certainly EF is prime or not. E ^ p p First let it be prime; therefore prime numbers A, B, C, EF are found more in number than A, B, C. But now let EF not be prime; therefore it is measured by some prime number; let it be measured by the prime number G ; I say that G is the same with neither of the numbers A, B, C For if G be the same with one of the numbers A, B, C, the numbers A, B, C, measure DE; therefore also G will measure DE; and it also mea- sures EF; and therefore G, being a number, will measure the remainder, the unity DF, which is absurd. Therefore G, is not the same with one of the numbers A, B, C, and it is supposed prime. Therefore prime num- bers. A, B, C, G, have been found more than the proposed multitude A, B, C, which was to be demonstrated." (See Book IX. Prop. 20.) It is easy to perceive that if IST and N -|- 1 be both composite numbers and a, b, c,be the prime factors of N, the factors of N -f- 1 must be not only all different from either one of the factors of N, but also less in numier and greater in magnitude. For every composite number is the least common multiple of its prime factors. Now, no two such multiples whose differ- ence is 1 only, can have any common prime factors or measures. And if the number of the factors of N -(- 1 ^^ not less in number than those of N, their product must necessarily exceed N -|- 1. To prove that prime numbers are infinite, a demonstration will be here added which is original and probably new. It is founded upon the con- sideration that the process used for sifting out the composite, from the prime and composite numbers, must necessarily always leave some that are prime, whatever be the number of the terms of the natural series taken to be sifted. The demonstration is the same in principle as that which may be used to prove the infinite divisibility of matter or of any mathe- matical quantity. If from any quantity which may be represented as a unit (1,) one-half he taken, then one-half of the remainder, then one-half of the second and «very successive remainder, ad infinitum, it is evident that there must al- ways be a remainder; so that the unit or whole quantity can never be ex- hausted. The parts thus taken out are truly represented by the series, T' 2^' i^W^i ^^- ^^ ^"™ °^ which will ever approach but never reach a unit or the whole quantity. Now the process of sifting out the composite, for the purpose of ascer- taining the prime numbers of any series, is, in effect, to take out of the entire number of terms, first all that are divisible by the least prime num- ber, then from the remainder all that are divisible by the next greater one, and M) on, using for a divisor every prime number not greater than the square root of the whole number of the terms of the given series. 216 APPENDIX. If we include in the mass to be sifted all the terms, odd and even, of the given series, we must begin the process with the divisor 2, and proceed with every other one of the prime numbers to the prescribed limit. But, if we first put into the sieve only those terms of the series which end in 1, 3, 7 or 9, among which all the primes, except 2 and 5, are contained, we must then omit the use of the numbers 2 and 5 as divisors, which measure six-tenths of the terms of every series whose last term ends in 0, and be- gin with the divisor 3, and then proceed with 7, 11, 13, &c. The number of terms sifted out by the divisor 3, is one-third of the whole, inclusive of itself, known to be prime; the number sifted by 7 is one-seventh of the remainder, inclusive of itself, a prime number, &c. Now it is evi- dent that this process, terminated at its proper limit, will not exhaust all the terms of the given series, although there are included in the sum of all the quotients as many prime numbers as have been used as divisors; and the last remainder will represent only those prime numbers of the series which are of greater magnitude than the greatest prime used as a divisor. The different quotients, in orderly succession and their sum, may be ex- pressed by numerical fractions, as parts of a unit, thus : i + ^ of f + iV of ^ of f + tV of H of 4 of f, &c., or 1 1^1-3 1^ 1-62 1^ 1-10-6-3 „ i _i_ ^ _1_ ii _1_ iH2. B, 3 ~r7-3"ril-7-3"ri311-7-3' 3 ~r 21 ~r 231 ~r 3003' '*•*'• The sum of this series, how far soever it be extended, can never be equal to a unit. The truth of the above formula may be tested by a table of prime numbers. For if the number of terms assumed to be sifted be an even number and at the same time a multiple of 3, the result will be true exactly or very nearly so. Let it be proposed to find the number of composite and prime numbers in the first three centuries. The numbers ending in 1, 3, 7, 9 are 120 in the three centuries. As ^300 exceeds 17, this number is the last divisor to be used. Whole number, 120 divided by 3 s= 40 = 39 composite and 1 prime. 40 First remainder 80 divided by 7 = 11 = 10 composite and 1 prime. 11 2nd remninder 69 divided by 11 = 6 = 5 composite and 1 prime. 6 3rd remainder 63 divided by 13 = 4 = 3 composite and 1 prime. 4 4tb remainder 59 divided by 17 = 3 = 3 composite and 1 prime. 64 59 5 Last remainder 56 The number remaining in the sieve is 56. And 56 -\- 5^ 61, which, exclusive of 3 and 5, is the exact number of primes in the first three cen- turies. It is certain therefore, that prime numbers are infinite.