Ex. 1. What is the greatest common measure of 18, 30 and Ans. 2 X 36. We see that 2 and 3 are factors common to all the numbers, and, furthermore, they are the only common factors; hence their product, 2 × 3 = 6, is the greatest common measure of the given numbers. 2. What is the greatest common measure of 60, 72, 48 and Ans. 22 X 3 = 12. 84? 3. What is the greatest common measure of 30, 90, 120, 210 and 60? Ans. 2 X3 X5 = 30. 4. Find the greatest common measure of 25, 75, 90, 85, 100, 65, 125 and 250. Ans. 5. 5. Find the greatest common measure of 42, 63, 105, 147, 189 and 168. 6. Find the greatest common measure of 72, 120, 144, 168 and 48. (a) When the given numbers are not readily resolved into their prime factors, their greatest common measure may be more easily found by RULE 2. Divide the greater of two numbers by the less, and, if there be a remainder, divide the divisor by the remainder, and continue dividing the last divisor by the last remainder until nothing remains; the last divisor is the greatest common measure of the two numbers. If more than two numbers are given, find the greatest measure of two of them, then of this measure and a third number, and so on until all the numbers have been taken; the last divisor will be the measure sought. 7. What is the greatest common measure of 14 and 20? OPERATION. 14) 20 (1 Ans. 2. 6) 14 (2 2) 6 (3 111. Before explaining this operation, four principles may be stated, viz.: (a) Every number is a measure of itself (101, Note 3). (b) If one number measures another, the 1st will measure any multiple of the 2d; thus, if 3 measures 12 it will measure 5 times 12, or any number of times 12. (c) If a number measures each of two numbers, it will measure their sum and also their difference; thus, since 6 is contained in 30 five times, and in 12 twice, in 30 + 12 = 42, it will be contained 5+2 = 7 times, and in 30 - 12 = 18, it will be (d) Not only will the greatest common measure of two numbers measure their difference, but, unless one of the numbers is a multiple of the other, it will also measure the remainder, after one of the numbers has been taken from the other, as many times as possible; thus, the greatest measure of 6 and 22 will measure 22 - 3 × 6 = 4. 112. It may now be shown, 1st, that 2 is a common measure of 14 and 20, and, 2d, that it is their greatest common measure. 1st. 2 measures 6, .. (111, b) 2 measures 6 × 2 = 12, and (111, c) 2 measures 2+12=14; again, since 2 measures 6 and 14 (111, c) it measures 6 + 14 = 20; i. e. 2 measures 14 and 20. 2d. The greatest measure of 14 and 20 (111, c) must measure 2014: - 6, .. it cannot be greater than 6; again, the greatest measure of 6 and 14 (111, d) must measure 14 — 6 × 2 = 2,.. the greatest common measure of 14 and 20 cannot exceed 2, and, as it has been previously shown that 2 is a measure of 14 and 20, it is their greatest measure. A similar explanation is applicable in all cases. 113. It will be seen that, in finding the common measure of 14 and 20, we are led to find the measure of 6 and 14, then of 2 and 6; i. e. in any example, we seek to find the measure of the remainder and divisor, then of the next remainder and divisor, and so on, until the greatest measure of the last remainder, and the divisor which gave that remainder, is found, and this measure will be the greatest common measure of the two given numbers; thus, the question becomes more and more simple as each successive step is taken in the operation. 8. What is the greatest common measure of 27088 and 39912? Ans. 8. 9. Find the greatest common measure of 437437 and 2018835. Ans. 91. 10. Find the greatest common measure of 16, 24 and 36. First find the measure of 16 and 24, viz., 8, and then find the measure of 8 and 36; or, first find the measure of 24 and 36, viz., 12, and then of 12 and 16; or, we might first find the measure of 16 and 36, and then of that measure and 24. 11. Find the greatest common measure of 9360, 437437 and 2018835. Ans. 13. 12. Find the greatest common measure of 1269729, 405405 and 5816907. 13. What is the greatest common measure of 8 and 15? Ans. 1. 14. What is the greatest common measure of 8, 12 and 33? 15. Find the greatest common measure of 1181, 2741 and 3413. PROBLEM 5. 114. To find the least common multiple of two or more numbers, RULE 1.—Resolve each number into its prime factors, and the continued product of the highest powers of all the different prime factors contained in the given numbers, will be the multiple sought. Ex. 1. What is the least common multiple of 24, 36 and 20? Ans. 2 × 2 × 2 × 3 × 3 × 5 = 23 × 32 × 5 360. Since 360 contains all the factors of 24, 36 and 20, respectively, it, evidently, is divisible by each of those numbers. It, also, is evident that no number less than 360 will contain 24, 36 and 20, for if one of the 2's in the common multiple were omitted, it would not contain 24; if one of the 3's, it would not contain 36; and if the 5 were omitted, it would not contain 20. 2. What is the least common multiple of 6, 8, 12, 18 and 24? Ans. 28 X 32 = = 72. 3. Find the least common multiple of 48, 96, 144 and 192. 4. Find the least common multiple of 33, 44 and 55. 5. Find the least common multiple of 3, 8, 27, 24, 54, 48, 90 and 45. 6. Find the least common multiple of 18, 28, 56, 64 and 72. (a) The above rule is always applicable, but the same end may sometimes be more easily attained by RULE 2.—Having set the given numbers in a line, divide by any PRIME number that will divide two or more of them, and set the quotients and undivided numbers in a line beneath; proceed with this line as with the first, and so continue until no two of the numbers can be divided by any number greater than one; the continued product of the divisors and numbers in the last line will be the multiple sought. This rule may be illustrated by the example already employed in explaining the first rule, viz., What is the least common multiple of 24, 36 and 20? OPERATION. Ans. 2 X2 X3 X2 X3 X5 = 360. 2)24, 36, 20 If the process by the 1st rule be examined it will be seen that the factor 2 is found 7 times in the given numbers, and as 2 is taken but 3 times in finding the multiple, it is rejected 4 times. By the 2d rule, also, 2 is rejected 4 times, viz., twice in the 1st division by 2 and twice in the 2d division by 2. The learner may think 2 is rejected 3 times in each of the two first divisions, but he must remember that the divisor, 2, is retained as a factor in the common multiple in each instance. 2, 3, 5 Similar remarks are applicable to all rejected factors in like examples, ... the two rules give identical results. NOTE. The principle, which is the same in the two rules, is most readily perceived by the first operation. 7. What is the least common multiple of 5, 16, 24, 32 and 48? Ans. 25 X3 X5 = 480. |