An Even Number is one whose right hand figure is 0, 2, 4, 6, or 8. An Odd Number is one whose right hand figure is 1, 3, 5, 7, or 9. A Composite Number is one that is composed of two or more factors. (25 and 28.) A Prime Number is one that has no factors except itself and unity. Numbers are prime to each other that have no common factor. Thus, 8 and 15, though composite numbers, are prime to each other, for 82 X2 X2, and 15=3 X 5. A Prime Factor of a number is a prime number that will divide it without a remainder. Thus, 2, 3, and 5, are prime factors of 30. 1. Is 20 an integer? Why? Is 18? Why? Is 3.8? Why? Is 7? Why? Is? Why? 2. Is 364 an even or an odd number? Is 497? 876? - 645? 3? 230? 3. Is 25 a composite or a prime number? Is 32? 63? 51? 37? 19? 22? 4. Are 9 and 12 prime to each other? Are 10 and 12? 7 and 15? 16, 24, and 25? 5. What are the prime factors of 6? Of 8? 9? 10? 11? 12? 15? 18? 23? 45? RULE FOR FINding all the PRIME FACTORS OF A NUMBER. Divide the number by any one of its prime factors; then that quotient by another, and so on till the quotient is a unit. The several divisors are all the prime factors of the number. 7. What are the prime factors of 1260? Ans. 2, 2, 3, 3, 5, 7. We see that a number is equal to the product of all its prime factors; for 2×2×3×3×5 X7=1260. 21260 2 630 3 315 5 105 7 21 3 3 1 NOTE. When a factor is repeated, it may be written but once, by placing a small figure above it at the right hand, called an index, to indicate how many times it is used as a factor. Thus, instead of writing 2 X2 X2 X3 X 3 X 5 = 360, we may write 23 X 32 X 5 = 360; 5X5X5X5×2×2×3× 3 × 3 = 67500, may be written 51 X 22 X 3367500. 8. What are the prime factors of 35? Of 48? Of 60? Of 72? 84? 275? 864? 1084? 35952? The following truths will aid the pupil in finding the prime factors and other divisors of numbers. 2 will divide all even numbers. 3 will divide all numbers the sum of whose figures is divisible by 3. Thus, 684 is divisible by 3, because 6+8+4=18 is divisible by 3. 4 will divide any number whose two right hand figures are divisible by 4. will divide any number whose right hand figure is 0 or 5. 6 will divide any even number that is divisible by 3. 8 will divide any number whose three right hand figures are divisible by 8. 9 will divide any number the sum of whose figures is divisible by 9. 10 will divide any number whose right hand figure is 0. 11 will divide any number in which the sum of the figures in the odd places is equal to the sum of the figures in the even places; or in which their sums differ by a number which can be divided by 11 without a remainder. Thus, 85426 is divisible by 11, because 8+4+6 =5+2+11. So is 63561894; because 6+5+1+9=3+6+ 8+4. 12 will divide any number which is divisible by 3 and 4, because A number that contains two or more factors that are prime to each other, is divisible by the product of those factors. 7, 11, and 13, are factors in numbers of 4 places in which two similar figures enclose two naughts; as 1001, 5005, 10010, 50050, &c. For a more extended list of factors, with their signs or marks of recognition, see a little work entitled "The Plain Calculator, by Lewis Joerres, Professor of Mathematics from Prussia." By what numbers is 348 divisible? Why? By what is 624 divisible? 24156? 463320? Why? 72. MEASURE. COMMON MEASURE. GREATEST COMMON MEASURE. A Measure of any number is a number that will divide it without a remainder. Thus, 3 is a measure of 3; of 6; 8 is a measure of 8, 16, &c. NOTE. A number is said to measure another when it will divide it without a remainder. Thus, 8 measures 32. What number is a measure of 24? What other? What other? A Common Measure of two or more numbers is a number that will measure each of them. Thus, 3 is a common measure of 6, 18, and 24; 4 is a common measure of 24, 32, 48. The Greatest Common Measure of two or more numbers is the greatest number that will measure them. Thus, 3 is a common measure of 6, 18, and 24; so is 2; but their greatest common measure is 6. What is the greatest common measure of 4 and 6? Of 6 and 9? Of 12 and 18? The common measure of two or more numbers contains no factors but those which are common to all the numbers; and their greatest common measure contains all those common factors, and no others. Hence the RULE FOR FINDING THE GREATEST COMMON MEASURE OF TWO OR MORE NUMBERS. Find all the prime factors of each of the numbers; then the product of all the factors, that are common to all the numbers, will be the great est common measure. 1. What is the greatest common divisor of 18, 24, 42, and 54? The only factors that are common to all the numbers are 2 and 3; therefore, the greatest common measure is 6. 2. What is the greatest common measure of 4, 8, 10, and 12? Of 12, 15, 21, 27? 3. What is the greatest common measure of 3, 9, 14, 18? Of 5, 6, 8, and 10? Of 28 and 54? 4. Find the greatest common measure of 42, 56, and 112. Of 84 and 120. 5. What is the greatest common measure of 136 and 352? Of 72 and 162? ANOTHER RULE. If there are but two given numbers, divide the greater by the less, and if there is no remainder, the divisor is the greatest common measure. If there is a remainder, divide the last divisor by it, and so on till nothing remains. The last divisor is the greatest com mon measure. Find the greatest common measure of 153 and 162. 153) 162 (1 153 9) 153 (17, the greatest common measure. 153 6. What is the greatest common measure of 184 and 232? Of 36 and 90? 7. Find the greatest common measure of 42, 56, and 91. First find the greatest common measure of 42 and 56, which is 14; and then of 14 and 91. Do the 2d, 3d and 4th, by this rule. 73. MULTIPLE. COMMON MULTIPLE. LEAST COMMON MULTIPLE. A Multiple of a number is a number that can be measured by it. Thus, 12 is a multiple of 6. What other number is a multiple of 6? Why? What other? What other? A Common Multiple of two or more numbers is a number that can be measured by each of them. mon multiple of 2, 3, 4, and 6. Thus, 24 is a com 1. Find a common multiple of 3, 4, and 8; another; another. 2. Find a common multiple of 5, 6, 10, and 15; of 2, 3, and 7. The Least Common Multiple of two or more numbers is the least number that can be measured by them. Thus, 24 is a common multiple of 3, 4, and 6; but their least common multiple is 12. 3. What is the least common multiple of 2, 3, and 4? Of 4, 5, and 6? Of 3, 6, and 8? Of 4, 5, and 10? Of 8 and 12? Of 12, 15, and 30? RULE FOR FINDING THE LEAST COMMON MULTIPLE OF TWO OR MORE NUMBERS. Every number is a multiple of itself; and since (71) any number is equal to the product of all its prime factors, any number that contains all the prime factors of a number must be a multiple of that number. For example, if 2 X 2 X 3 is a multiple of 12, any number of times 2x2 x3 must be a multiple of 12. Find the least common multiple of 15 and 18. 15=3×5; 1832 x 2. Any number that contains all these factors is a common multiple of 15 and 18, whether it contains other factors or not; and the number that contains these factors but once, and contains no other factors, is the least common multiple of 15 and 18. 2 × 32 × 5, 90, contains them all once, and only once; 90 is therefore the least common multiple of 15 and 18. 4. What is the least common multiple of 9, 11, 15, and 24? 9 32; 11 is a prime number; 15-3x5; 24-23× 3. 23 X3 X5 X 11=3960, Ans. = 5. What is the least common multiple of 18 and 35? 18=32 × 2; 35=7×5; no factor being common to both numbers, the common multiple must be the product of all of them. 32 X2 × 5 × 7 = = 810, Ans. 18 x 35 From these examples we deduce the following RULE. If the numbers are prime to each other, take the product of them all for the least common multiple. If they are not prime to each other, take the product of all the prime factors contained in the numbers, using each factor the greatest number of times it occurs in either of the given numbers. 6. What is the least common multiple of 6, 8, and 10? Of 5, 12, and 15? 4, 9, and 25? 7, 21, 25, and 35? 7. Find the least common multiple of 12, 16, and 18. 2, 3, 4, 5, and 6. Of 45, 72, and 84. Of 35, 56, and 63. CANCELLATION. 74. If, in division, both the dividend and divisor be multiplied or divided by the same number, the product or quotient will not be altered. Thus, 24 divided by 6 gives the same quotient as 48 divided by 12, or 72 divided by 18, or 12 divided by 3, or 8 divided by 2. Therefore, when division is to be performed, if the divisor and dividend have a common factor, the operation may often be shortened by cancelling or rejecting that factor, and performing the work without it. ΣΤ - 420 For example: suppose 12 times 35 is to be divided by 21. The work may be expressed thus: 1235 20. By separating 35 and 21 into their prime factors, it will be ex12 X 5X7 pressed thus, 3X7 420- 20. But as 7 is a factor of both the dividend and divisor, it may be cancelled from each, = = 60 = 20. by drawing a line across it, thus, 12 × 5 × 7 tors, and the expression will be 4X3X5X7 ; by cancelling The number 12 may, in like manner, be separated into fac |