FIRST OPERATION. SECOND OPERATION. 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. Resolving the numbers into their 24 2 X 2 X 2 X 3 prime factors, thus, 24 = 2 X 2 X 2 88 2 X 2 X 2 X 11. X 3, and 88 = 2 X 2 X 2 X 11= 88, we find the factors 2 x 2 x 2 2 x 2 x 2 = 8. Ans. 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 X 2 X 2= 8, the greatest common divisor required. The same result may be obtained by a sort of trial process, as by the second operation. It is evident, since 24 cannot be ex actly divided by a number greater than 2 4) 88 (3 itself, if it will also exactly divide 88, 7 2 it will be the greatest common divisor sought. But, on trial, we find 24 will 16) 24 (1 not exactly divide 88, there being a re16 mainder, 16. Therefore 24 is not a common divisor of the two numbers. 8)16 (2 We know that a common divisor of 16 16 and 24 will, also, be a common di visor 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. The last method may be often contracted, if there should be observed to be any prime factor 24) 88 (3 in a remainder which is not common to the pre7 2 ceding divisor, by canceling said factor. Thus, in the third operation, the factor 2 being found 16 in the remainder 16 once more than in the di8) 2 4 (3 visor 24, we cancel one 2 from 16, and, having 24 left the composite factor 8, we divide 24 by that factor. There being no remainder, 8 is the greatest common divisor, as before obtained. THIRD OPERATION. Or, 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. 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 divisor. 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. 4. What is the greatest common measure of 57 and 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 ? Ans. 1. 8. What is the greatest common measure of 47 and 478 ? 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 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 8= 2 X 2 X 2 numbers into their 16= 2 X 2 X 2 X 2 prime factors, we 24 2 X 2 X 2 X 3 find their differ32 2 X 2 X 2 X 2 X 2 ent prime factors 44 2 X 2 X 11 to be 2, 3, and 11. The greatest 2 X 2 X 2 X 2 X 2 X 3 X 11 = 1056 Ans. number of times the 2 occurs as a factor in any of the given numbers is 5 times; the greatest num FIRST OPERATION. SECOND OPERATION. 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). Having arranged the 2)8 16 24 3 2 44 numbers on a horizon tal line, we divide by 2) 4 8 12 16 2 2 2, a prime number that 2) 2 4 6 8 11 will divide two or more of them without a re2)1 2 3 4 11 mainder, and write the 1 1 3 2 11 quotients in a line be low; and we continue 2 X 2 X 2 X 2 X2 X3 X11 = 1056 Ans. to divide by a prime number as before, till 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. Thus, in the third operation, 4) 8 16 24 32 44 8 being a factor of several of the numbers, and 16 being a 8 11 factor of one other number, we 3 4 11 cancel them; and observing that 4 is the greatest common 4X2 X3 X4X11=1056 Ans. divisor of the remaining num bers, 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. The fourth operation exhibits $ 16 2 4 3 2 44 a process yet more contracted. The 8 and 16 being factors each 4 11 of one or more of the other num24 X 4 X 11 1056 Ans. bers, 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 the two remaining numbers the largest factor each has in common with the 24, by dividing each of them by the greatest common divisor THIRD OPERATION. 2) 6 FOURTH OPERATION. between it and 24, and write the result beneath. The numbers in the lower line having no factor in common, we carry the process no further. The continued product of the number cut off by the numbers in the lower line gives 1056, the least common multiple, as by the other methods. In this instance we cut off the 24, but either the 32 could have been separated from the rest, or the 44 cut off, and the needless factors striken out with like result. If, however, we had cut off the 44, the numbers placed in the second line would have contained factors common to each other, so that it would have been necessary in that line to have cut off and stricken out factors as before. The reason for this abridged process is, that by the separating off, and by the striking out of factors, we get rid, in an expeditious way, of the factors not required to form the least common multiple sought. Rule 1. Resolve the given numbers into their prime factors. The product of these factors, taking each factor only the greatest number of times it occurs in any of the numbers, will be the least common multiple. Or, RULE 2.-— Having arranged the numbers on a horizontal line, cancel such of them as are factors of any of the others, and separate some convenient one from the rest. Reject from each of the numbers remaining the greatest factor common to it and that number, and write the result in a line below. Should there be in the second line numbers having factors in common, proceed as before ; and so continue until the numbers written below are prime to each other. The continued product of the number or numbers separated from the others with those in the last line will be the least common multiple. NOTE 1. Some give a preference to the following rule for finding the least common multiple: Having arranged the numbers on a horizontal line, divide by such a prime number as will exactly divide two or more of them, and write the quotients and undivided numbers in a line beneath. So continue to divide until the quotients shall be prime to each other. Then the product of the divisors and the numbers of the last line will be the least common multiple. NOTE 2.— The least common multiple of two or more numbers that are prime to each other is found by multiplying them together (Art. 202). NOTE 3. - When a single number alone is prime to all the rest, it may be separated off, and used only as a factor of the least common multiple sought. NOTE 4. When the least common multiple of several numbers, and all the numbers except one, which is prime to the others, are given, to find the unknown number, divide the least common multiple given by that of the known numbers (Art. 202). EXAMPLES. 2. What is the least common multiple of 3, 13, 37, and 91. 3. What is the least common multiple of 9, 14, 30, 35, and 47 ? Ans. 29610. |