DEFINITIONS. 129. A Common Divisor of two or more numbers is an exact divisor of each of them. Thus, 6 is a common divisor of 12, 24, 48; 8 of 16, 24 and 64. 130. The Greatest Common Divisor of two or more numbers is the greatest number that is an exact divisor of each of them. Thus, 24 is the greatest common divisor of 24 and 48. 131. When numbers have no common divisor they are said to be Prime to each other. Thus, 7, 8 and 9 are prime to each other. A common divisor is sometimes called a common measure and the greatest common divisor the greatest common measure. 132. PRINCIPLE.-The greatest common divisor of two or more numbers is the product of all their common prime factors. 1. What is the greatest common divisor of 45, 60, and 75? 1ST PROCESS. 45=3X3X5 60=2×2×3×5 75=3X5×5 ,3X5=15 ANALYSIS.-Since the greatest common divisor is equal to the product of all the prime factors common to the given numbers, we separate the numbers into their prime factors. The only prime factors common to all these numbers are 3 and 5. Hence their product, 15, is the greatest common divisor of the given numbers. 2D PROCESS. 345 60 75 25 5 15 20 3 4 5 ANALYSIS.-3 will divide each of the given numbers, and is therefore a factor of the greatest common divisor. 5 will divide each of the resulting quotients and is therefore a factor of the greatest common divisor. The quotients 3, 4, and 5, have no common divisor; therefore 3 and 5 are the only factors of the greatest common divisor, 15. 3X5=15 RULE.-Separate the numbers into their prime factors and find the product of all the common factors. Or, Divide the numbers by any common divisor, the resulting quotients by another common divisor, and so continue to divide until quotients are obtained that have no common divisor. The product of the divisors will be the greatest common divisor. 133. When the numbers can not be factored readily, the following method is employed: 1. What is the greatest common divisor of 35 and 168? PROCESS. 35) 168 (4 28) 35(1 7)28(4 ANALYSIS.-The greatest common divisor can not be greater than the smaller number; therefore 35 will be the greatest common divisor if it is exactly contained in 168. By trial it is found that it is not an exact divisor of 168, since there is a remainder of 28. Therefore 35 is not the greatest common divisor. Since 168 and 140, which is 4 times 35, are each divisible by the greatest common divisor, their difference, 28, must contain the greatest common divisor; therefore the greatest common divisor can not be greater than 28. 28 will be the greatest common divisor if it is exactly contained in 35; since if it be contained in 35, it will be contained in 140, and in 28 plus 140, or 168. By trial we find that it is not an exact divisor of 35, for there is a remainder of 7. Therefore 28 is not the greatest common divisor. Since 28 and 35 are each divisible by the greatest common divisor, their difference, 7, must contain the greatest common divisor; therefore the greatest common divisor can not be greater than 7. 7 will be the greatest common divisor if it is exactly contained in 28; since if it be contained in itself and 28, it will be contained in their sum, 35, and also in 168, which is the sum of 28 and 4 times 35, or 140. By trial we find that it is an exact divisor of 28. Hence 7 is the greatest common divisor. RULE.-Divide the greater number by the less and if there be a remainder divide the less number by it, then the preceding divisor by the last remainder, and so on, till nothing remains. The last divisor will be the greatest common divisor. If more than two numbers are given, find the greatest common divisor of any two, then of this divisor and another of the given numbers, and so on. The last divisor will be the greatest common divisor. Find the greatest common divisor of 20. In a village some of the walks are 56 inches wide, Some 70 inches, and others 84 inches. What is the width of the widest flagging that will suit all the walks? 21. A merchant has 60 pounds of tea of one kind, 75 pounds of another, and 100 pounds of another, which he wishes to put up in the largest possible equal packages without mixing the different kinds. How many pounds should be put in each package? 22. Mr. A. has 324 acres of land in one farm and 78 acres in another. He wishes to divide these into the largest possible fields of equal size. How many fields will there be, and how many acres in each field? MULTIPLES. 134. 1. What numbers less than 25 will exactly contain 4? 5? 6? 2. What numbers less than 25 will exactly contain both 4 and 6? 3. Name some numbers that are exactly divisible by 5. By 4. By both 5 and 4. 4. Name some numbers that are exactly divisible by 2. By 3. By 4. By 2 and 4. 5. What is the smallest number that is exactly divisible by each of the numbers 2, 3, and 4? 6. What is the least number that will contain 10 and 15? 7. What common prime factors have 10 and 15? What factor occurs in 10 that does not in 15? What factor is found in 15 that is not found in 10? 8. What are all the different prime factors of 10 and 15? 9. How may the least number that will contain 10 and 15 be formed from their prime factors? What is the least number that will exactly contain DEFINITIONS. 135. A Multiple of a number is a number that will exactly contain it. A multiple of a number is obtained by multiplying the given number by some integer. 136. A Common Multiple of two or more numbers, is a number that will exactly contain each of them. 137. The Least Common Multiple of two or more numbers, is the least number that will exactly contain each of them. 138. PRINCIPLE.-The least common multiple of two or more numbers is equal to the product of all the prime factors of the numbers, and no other factors. WRITTEN EXERCISES. 139. 1. Find the least common multiple of 30, 28 and 60? 1ST PROCESS. 30=2X3X5 28=2×2×7 2×2×3×5X7=420 ANALYSIS.-Since the least common multiple is equal to the product of all the different prime factors of the numbers and no other factors, (Prin.) the numbers must be separated into their prime factors, and the product of all the different prime factors found. The prime factors of 60, the largest number, are 2, 2, 3 and 5. 28 contains a factor, 7, which is not found in 60. 60 contains all the factors of the other number, 30. Therefore all the different prime factors of the given numbers are 2, 2, 3, 5 and 7, and their product, 420, is the least common multiple. RULE.-Separate the given numbers into their prime factors. Find the product of all the different prime factors, using each factor the greatest number of times it occurs in any of the given numbers. 2 |