81. Divisibility. A number is said to be divisible by any one of its factors. 82. Tests of Divisibility. A number is divisible by 2 if its last digit is divisible by 2. Zero is here regarded as divisible by 2. Thus, 364, 748, 196, 48780 are divisible by 2. A number is divisible by 4 if the last two of its digits represent a number which is divisible by 4. Thus, 324, 25632, 70976, 134568 are all divisible by 4. A number is divisible by 8 if the last three of its digits represent a number which is divisible by 8. Thus, 7144, 245632, 19248, 3749672 are all divisible by 8. All numbers which end in 5 or 0 are divisible by 5. All numbers which end in 0 are divisible by 10. A number is divisible by 3 if the sum of its digits is divisible by 3. Thus, 432423 is divisible by 3 because 4 + 3 + 2 + 4 + 2 + 3 divisible by 3. = 18 is A number is divisible by 9 if the sum of its digits is divisible by 9. A number is divisible by 6 if it is even and is divisible by 3. Thus, 48732 is divisible by 6, while neither 3442 nor 753 is divisible by 6. A number is divisible by 12 if it is divisible by both 3 and 4. 83. Divisors. If a given number is divisible by a number, then this second number is called a divisor of the given number. ORAL EXERCISES State which of the numbers 1, 2, 3, 4, 5, 6, 8, 10, 12, if any, are divisors of each of the following numbers : 84. Factoring. Factoring a number consists in finding all the prime factors of the number. The product of all the prime factors of a number is the number itself. Thus, the prime factors of 5 are 1 and 5, and 1 × 5 = 5. The prime factors of 6 are 1, 2, and 3, and 1 × 2 × 3 = 6. Sometimes one number is used as a factor several times. Thus, 2 is used as a factor 3 times in 8, because 2 × 2 × 2 = 8. 85. Common Factors. ·If a number is a factor of each of two or more numbers, it is said to be a common factor of these numbers. Thus, 2 is a common factor of 6, 8, and 10. 3 is a common factor of 9, 18, 21; and 7 is a common factor of 28 and 42. 86. Greatest Common Divisor. The greatest number which is a common factor of two or more numbers is called the Greatest Common Divisor (G. C. D.) of these numbers. The G. C. D. of two or more numbers is the product of all the prime common factors of these numbers. Example. Find the G. C. D. of 32, 64, 96, 128. The work is conveniently arranged thus: Divide each number by the common factor 2, obtaining 16, 32, 48, 64 as the quotients. 2 is a common factor of the quotients. Continuing in this manner, we finally get 1, 2, 3, 4 as quotients, and these have no common factor except 1. Hence, the G. C. D. 2 X 2 X 2 × 2 × 2 32. = = EXERCISES 1. Find all prime factors of: 6, 8, 10, 12, 16, 18, 21, 24, 32, 36, 42. 2. Find all prime common factors of 18, 24, 36, 42. Find the G. C. D. of each of the following sets of numbers: 3. 30, 42, 64, 72 4. 16, 96, 108, 124 5. 14, 21, 49, 63 6. 256, 768, 508, 1164 9. 17, 51, 85, 102 10. 9, 33, 117, 141 11. 57, 95, 152, 190 12. 195, 270, 450 13. 75, 125, 340 14. 39, 52, 91, 104 87. A General Principle on Common Factors. - If a number is common factor of two numbers, it is a factor of their sum and also of their difference. Thus, 8 is a common factor of 24 and 56 and therefore of 56+ 24 = 80 and of 5624 = 32. This principle may be used in finding common factors. Example. Find the G. C. D. of 1292 and 1615. Solution. 1615 1292 323 4 323)1292 1292 If these numbers have a common factor, it is a factor of their difference, that is, of 323. By dividing we find 323 to be a factor of 1292 and of 1615. Hence 323 is the required G. C. D. The general application of this principle is too complicated for this book and is very seldom if ever required in business practice. 88. Multiples. The product obtained by multiplying a given. number by an integer is called a multiple of the given number. Thus, 6, 12, 18, 24 are multiples of 6. These numbers are also multiples of 2 and 3. Every integer is a multiple of 1. 89. Rule on Multiples. If one of two numbers is a multiple of the other number, then every multiple of the first number is a multiple of the second. That is, every multiple of 16 is a multiple of 2, 4, and 8; and every multiple of 60 is a multiple of 2, 3, 5, 10, 20, and 30. 90. Common Multiples. - A number which is a multiple of each of two or more numbers is called a common multiple of these numbers. That is, 30 is a common multiple of 5, 6, 10, 2, 3. ORAL EXERCISES Find two common multiples of each of the following sets of numbers: 91. Least Common Multiples. The smallest number which is a common multiple of two or more numbers is called the Least Common Multiple of these numbers. The Least Common Multiple is usually denoted by L. C. M. Thus the L. C. M. of 10, 15, 20 is 60, and the L. C. M. of 8, 16, 24 is 48. Finding the L. C. M. of numbers is of importance in dealing with fractions. Example. Find the L. C. M. of 18, 32, 48. The L. C. M. of 18, 32, 48 must be a multiple of 48. That is, it must contain the factors 2, 2, 2, 2, 3. Since the L. C. M. is a multiple of 32 it must contain an additional factor 2, and since it is a multiple of 18, it must contain an additional factor 3. Hence the L. C. M. of 18, 32, 48 is 2 × 2 × 2 × 2 × 2 × 3 × 3 = 288. This problem may also be solved as follows: 2)18, 32, 48 3)9, 16, 24 8)3, 16, 8 3, 2, 1 First divide each number by 2. Then divide 9 and 24 by 3, bringing down the 16. Finally divide 16 and 8 by 8, bringing down the 3. The L. C. M. is 2 × 3 × 8 × 3 × 2 = 288. The process of division ends when no two numbers below the last line contain a common factor. The L. C. M. is the product of the numbers below the last line and of the factors to the left. At each step divide by the largest number possible. In finding the L. C. M. of the numbers 18, 24, 36, 48, we need only to find the L. C. M. of 36 and 48, since 18 and 24 are contained in these. For the purpose of finding the L. C. M. of a set of numbers any number which is contained in one of the others may be rejected. EXERCISES Find the L. C. M. of each of the following sets of numbers. In each case, state what numbers may be rejected. MISCELLANEOUS WORK 1. Loads of grain were hauled to an elevator. Find the net weight in pounds if the gross weights and tares were as follows: In each of the following find by how much the first number exceeds the sum of the numbers below it. Copy the following from dictation. Find the differences by hori |