52. To find the greatest common measure of three or more numbers. RULE. Find the greatest common measure of the first two numbers; then the greatest common measure of the common measure so found and the third number; then that of the common measure last found and the fourth number, and so on. The last common measure so found will be the greatest common measure required. Ex. Find the greatest common measure of 16, 24, and 18 16) 24 (1 16 8) 16 (2 therefore 8 is the greatest common measure of 16 and 24. Now to find the greatest common measure of 8 and 18, 8) 18 (2 2) 8 (4 8 0 therefore 2 is the greatest common measure required. Reason for the above process. It appears from Art. (50) that every number, which measures 16 and 24, measures 8 also; therefore every number, which measures 16, 24, and 18, measures 8 and 18; therefore the greatest common measure of 16, 24, and 18, is the greatest common measure of 8 and 18. But 2 is the greatest common measure of 8 and 18; therefore 2 is the greatest common measure of 16, 24, and 18. 53. A COMMON MULTIPLE of two or more given numbers is a number which will contain each of the given numbers an exact number of times without a remainder. Thus, 144 is a common multiple of 3, 9, 18, and 24. The LEAST COMMON MULTIPLE of two or more given numbers is the least number which will contain each of the given numbers an exact number of times without a remainder. Thus, 72 is the least common multiple of 3, 9, 18, and 24. 54. To find the least common multiple of two numbers. RULE. Divide their product by their greatest common measure: the quotient will be the least common multiple of the numbers. Ex. Find the least common multiple of 18 and 30. Proceeding by the Rule given above, 18) 30 (1 18 12) 18 (1 12 6) 12 (2 12 0 therefore 6 is the greatest common measure of 18 and 30. 18 30 6 540 90 therefore 90 is the least common multiple of 18 and 30. Reason for the above process. 18=3 × 6, and 30=5 × 6. Since 3 and 5 are prime factors, it is clear that 6 is the greatest common measure of 18 and 30; therefore their least common multiple must contain 3, 6, and 5, as factors. Now every multiple of 18 must contain 3 and 6 as factors; and every multiple of 30 must contain 5 and 6 as factors; therefore every number, which is a multiple of 18 and 30, must contain 3, 5, and 6 as factors; and the least number which so contains them is 3 × 5 × 6, or 90. Now, 90=(3 × 6) × (5 × 6), divided by 6, = 18×30, divided by 6, = 18 × 30, divided by the greatest common measure of 18 and 30. 55. Hence it appears that the least common multiple of two numbers, which are prime to each other, or have no common measure but unity, is their product. 56. To find the least common multiple of three or more numbers. RULE. Find the least common multiple of the first two numbers; then the least common multiple of that multiple and the third number, and so on. The last common multiple so found will be the least common multiple required. Ex. Find the least common multiple of 9, 18, and 24. Proceeding by the Rule given above, Since 9 is the greatest common measure of 18 and 9, their least com mon multiple is clearly 18. Now, to find the least common multiple of 18 and 24. 18) 24 (1 18 6) 18 (3 0 therefore 6 is the greatest common measure of 18 and 24; therefore the least common multiple of 18 and 24 is equal to (18 × 24) divided by 6, 24 18 192 24 6432 72 therefore 72 is the least common multiple required. Reason for the above process. Every multiple of 9 and 18 is a multiple of their least common multiple 18; therefore every multiple of 9, 18, and 24 is a multiple of 18 and 24; and therefore the least common multiple of 9, 18, and 24 is the least common multiple of 18 and 24: but 72 is the least common multiple of 18 and 24; therefore 72 is the least common multiple of 9, 18, and 24. 57. When the least common multiple of several numbers is required, the most convenient practical method is that given by the following Rule. RULE. Arrange the numbers in a line from left to right, with a comma placed between every two. Divide those numbers which have a common measure by that common measure, and place the quotients so obtained and the undivided numbers in a line beneath, separated as before. Proceed in the same way with the second line, and so on with those which follow, until a row of numbers is obtained in which there are no two numbers which have any common measure greater than unity. Then the continued product of all the divisors and the numbers in the last line will be the least common multiple required. Note. It will in general be found advantageous to begin with the lowest prime number 2 as a divisor, and to repeat this as often as can be done; and then to proceed with the prime numbers 3, 5, &c. in the same way. Ex. Find the least common multiple of 18, 28, 30, and 42. therefore the least common multiple required =2×2×3×7 × 3 × 5=1260. Reason for the above process. 1 Since 18=2×3×3; 28=2×2×7; 30=2×3×5; 42=2×3×7; it is clear that the least common multiple of 18 and 28 must contain as a factor 2×2×3×3×7; and this factor itself is evidently a common multiple of 2 × 3 × 3, or 18, and of 2 × 2 × 7, or 28; now the least number which contains 2×2×3×3×7 as a factor, is the product of these numbers; therefore 2×2×3×3×7 is the least common multiple of 18 and 28: also it is clear that the least common multiple of 18, 28 and 30, or of 2×2×3×3×7 and 30, or of 2×2×3×3×7 and 2×3×5 must contain as a factor 2 × 2 × 3 × 3 × 7×5, and this factor itself is evidently a common multiple of 2× 3 × 3 or 18, 2 × 2 × 7 or 28, and 2 × 3 × 5 or 30; hence it follows as before that 2×2×3×3×7×5 is the least common multiple of 18, 28, and 30; again the least common multiple of 2×2×3×3×7 × 5 and 42, or of 2×2×3×3×7×5 and 2×3×7 must contain 2×2×3×3×7×5 as a factor, and this factor, as before, is evidently itself a common multiple of 18, 28, 30, and 42; now the least number which contains 2 × 2 × 3×3×7×5 as a factor, is the product of these numbers. Therefore this product, or 1260, is the least common multiple required. |