196. A common divisor of two numbers is a divisor of their sum, and also of their difference. Thus, 6, a common divisor of 12 and 18, is a divisor of their sum, 30, and of their difference, 6. 197. A common divisor of the remainder and the divisor is a divisor of the dividend. Thus, in a division having 8 for a remainder, 16 for divisor, and 24 for dividend, 8, a common divisor of 8 and 16, is also a divisor of the 24. 198. To find all the divisors common to two or more numbers. Ex. 1. Required all the common divisors of 45 and 135. Ans. 1, 3, 5, 9, 15, and 45. Resolving the giver numbers into their prime factors, we find they have of these 3, 3, and 5 in common, and these COMMON 3 × 3 × 5. prime factors with 1, and all the products we are able to form from them (Art. 194), give Ans. 1, 3, 5, 9, 15, and 45. all the common divisors required. When only the number of common divisors is required, it may readily be found by multiplying together the exponents, each increased by 1, of the different COMMON prime factors. (Art. 194.) EXAMPLES. 2. What are the common divisors of 51, 153, and 255 ? 3. Required the several common divisors of 180 and 360. Ans. 1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, and 180. 4. How many common divisors have 2025, 6075, and 8100? Ans. 15. 5. How many common divisors have 4500 and 9000? Ans. 36. THE GREATEST COMMON DIVISOR OR MEASURE. 199. The greatest common divisor or measure of two or more numbers is the greatest number that will divide each of them without a remainder. Thus, 4 is the greatest common divisor of 8, 12, and 16. 200. To find the greatest common divisor of two or more numbers. Ex. 1. Required the greatest common divisor or measure of 24 and 88. Ans. 8. 24 88 = = FIRST OPERATION. 2 X 2 X 2 X 3 2 X 2 X 2 × 11. 2 X 2 X 2 8. Ans. = Resolving the numbers into their prime factors, thus, 24 = 2 × 2 × 2 X 3, and 88 2 X 2 X 2 X 11 = 88, we find the factors 2 × 2 × 2 are common to both. Since only these common factors, or the product of two or more of such factors, will exactly divide both numbers, it follows that the product of all their common prime factors must be the greatest factor that will exactly divide both of them. Therefore, 2 × 2 X 28, the greatest common divisor required. The same result may be obtained by a sort of trial process, as by the second operation. SECOND OPERATION. 24) 88(3 72 16) 24 (1 8) 16 (2 It is evident, since 24 cannot be exactly divided by a number greater than itself, if it will also exactly divide 88, it will be the greatest common divisor sought. But, on trial, we find 24 will not exactly divide 88, there being a remainder, 16. Therefore 24 is not a common divisor of the two numbers. We know that a common divisor of 16 and 24 will, also, be a common divisor of 88 (Art. 197). We next try to find that divisor. It cannot be greater than 16. But 16 will not exactly divide 24, there being a remainder, 8; therefore 16 is not the greatest common divisor. As before, the common divisor of 8 and 16 will be the common divisor of 24 and 88 (Art. 197); we make trial to find that divisor, knowing that it cannot be greater than 8, and find 8 will exactly divide 16. Therefore 8 is the greatest common divisor required. THIRD OPERATION. 24) 88(3 72 16 The last method may be often contracted, if there should be observed to be any prime factor in a remainder which is not common to the preceding divisor, by canceling said factor. Thus, in the third operation, the factor 2 being found in the remainder 16 once more than in the di 8) 24 (3 visor 24, we cancel one 2 from 16, and, having left the composite factor 8, we divide 24 by that factor. There being no remainder, 8 is the greatest common divisor, as before obtained. 24 RULE 1. - Resolve the given numbers into their prime factors. The product of all the factors common to the several numbers will be the greatest common divisor. Or, RULE 2. Divide the greater number by the less, and if there be a remainder divide the preceding divisor by it, and so continue dividing until nothing remains. The last divisor will be the greatest common di visor. NOTE. When the greatest common divisor is required of more than two numbers, find it of two of them, and then of that common divisor and of one of the other numbers, and so on for all the given numbers. The last common divisor will be the greatest common divisor required. Another method is to divide the numbers by any factor common to them all; and so continue to divide till there are no longer any common factors; and the product of all the common factors will be the greatest common divisor required. EXAMPLES. 2. What is the greatest common divisor of 56 and 168? 285? Ans. 57. 5. What is the greatest common measure of 169 and 175? 6. What is the greatest common measure of 175 and 455? Ans. 35. 7. What is the greatest common divisor of 169 and 866? 8. What is the greatest common measure 478? Ans. 1. of 47 and Ans. 1. 9. What is the greatest common measure of 84 and 1068? Ans. 12. 10. What is the greatest common divisor of 75 and 165? Ans. 15. 11. What is the greatest common measure of 78, 234, and 468? Ans. 78. 12. I have three fields; one containing 16 acres; the second, 20 acres; and the third, 24 acres. Required the largest-sized lots, containing each an exact number of acres, into which the whole can be divided. Ans. 4 acre lots. 13. A farmer has 12 bushels of oats, 18 bushels of rye, 24 bushels of corn, and 30 bushels of wheat. Required the largest bins, of uniform size, and containing an exact number of bushels, into which the whole can be put, each kind by itself, and all the bins be full. LEAST COMMON MULTIPLE. 201. A common multiple of two or more numbers is a number that can be divided by each of them without a remainder; thus, 14 is a common multiple of 2 and 7. The least common multiple of two or more numbers is the least number that can be divided by each of them without a remainder; thus, 12 is the least common multiple of 4 and 6. 202. A multiple of a number contains all the prime factors of that number; the common multiple of two or more numbers contains all the prime factors of each of the numbers; and the least common multiple of two or more numbers contains only each prime factor taken the greatest number of times it is found in any of the several numbers. Hence, 1. The least common multiple of two or more numbers must be the least number that will contain all the prime factors of them, and none others. 2. The least common multiple of two or more numbers, which are prime to each other, must equal their product. 3. The least common multiple of two or more numbers must equal the product of their greatest common divisor, by the factors of each number not common to all the numbers. 4. The least common multiple of two or more numbers, divided by any one of them, must equal the product of those factors of the others not common to the divisor. 203. To find the least common multiple of two or more numbers. Ex. 1. What is the least common multiple of 8, 16, 24, 32, 44. Ans. 1056. Resolving the numbers into their prime factors, we find their different prime factors to be 2, 3, and 11. The greatest 2 X 2 X 2 X 2 × 2 × 3 × 11 1056 Ans. number of times = the 2 occurs as a factor in any of the given numbers is 5 times; the greatest num ber of times 3 occurs in any of the numbers is once; and the greatest number of times the 11 occurs in any of the numbers is once. Hence, 2, 2, 2, 2, 2, 3, and 11 must be all the prime factors necessary in composing 8, 16, 24, 32, and 44; and consequently, 1056, the product of these factors, is the least common multiple required (Art. 202). the divisor and remainders are all prime to each other. Then, these, since they include all the factors necessary to form the given numbers and no others, we multiply together for the required least common multiple, and obtain 1056, as before. The least common multiple of two or more numbers may be found generally by a process much shorter than either of the above methods, by canceling any number that is a factor of any other of the given numbers, and also by dividing the numbers by such a composite number as may be observed to be their common or greatest common divisor. THIRD OPERATION. 4)8 16 24 32 44 2) 6 8 11 3 4X2X3X4X11 4 11 Thus, in the third operation, 8 being a factor of several of the numbers, and 16 being a factor of one other number, we cancel them; and observing that 4 is the greatest common divisor of the remaining numbers, we divide them by it. We next divide by 2, as in the second operation. The numbers in the lower line then being prime to each other, we multiply them and the divisors together, and obtain 1056 as the least common multiple. FOURTH OPERATION. 8 16 24 32 44 The fourth operation exhibits a process yet more contracted. The 8 and 16 being factors each of one or more of the other numbers, we cancel them, as in the third operation. Of the remaining numbers we cut off 24 by a short vertical line from the rest as a factor of the least common multiple sought. We then strike out of 24 X 4 X 11 the two remaining numbers the largest factor each has in common with the 24, by dividing each of them by the greatest common divisor |