FINDING THE G. C. D. OF LARGE NUMBERS. When the G. C. D. is to be found of numbers which cannot be be readily factored, the following method is used: If the G. C. D. of two numbers is to be found, smaller; then divide the divide the larger by the divisor last used by the last remainder, and so on, until there is no remainder. The final divisor is the G. C. D. ILLUSTRATION: Find the G. C. D. of 592 and 333. A very common mistake made by pupils who are learning to find the G. C. D. of two or more large num bers, is this: They take the result obtained last for the G. C. D., instead of the final divisor. Notice whether or not your pupils are too observing to forget which is the G. C. D. To find the G. C. D. of three or more numbers, the above process is carried out with all of them. The G. C. D. of any two is found first. Next find the G. C. D. of the third number and the G. C. D. (already found) of the other two. Proceed in this way and the last divisor will be the G. C. D. of all the numbers. TO ILLUSTRATE: Find the G. C. D. of 3070, 2149, and 614. = EXPLANATION: Find the G. C. D of 3070 and 2149. It is 307. Find the G. C. D. of 307 and 614. It is 307. Then 307 is the G. C. D. of 3070, 2149, and 614. 307 G. C. D. of 3070, 2149, 614. Let the child try other examples like the above, until he is familiar with the process. He is just as liable to be called upon to find the G. C. D. of large numbers in real life, as he is the G. C. D. of small ones. Some numbers, as those given below, are found to be prime to each other only after performing the work. G. C. D. = 1)7 (7 LEAST COMMON MULTIPLE. PRESENT IN THIS WAY: Name a number of which 2 is a factor. Name one of which 3 is a factor. Of which 5 is a factor. Name a number which is exactly divisible by 5. (Child) 15. By 3. (Child) 9. We call 15 a multiple of 5. We call 9 a multiple of 3. Can you tell what a multiple is now, in your own words? Name a number that is exactly divisible by both 5 and 3. (Child) 15. (Child) 15. Another. (Child) 30. Another. (Child) 45. We call 15 a common multiple of 5 and 3. Name other common multiples of 5 and 3. What do you mean by common multiple? Name the least number that is exactly divisible by both 5 and 3. (Child) 15. We call 15 the Least Common Multiple of 5 and 3. Why? A multiple of a number is a number that is ex actly divisible by it. When we multiply a number we get a larger num ber, so a multiple of a number is always larger than the number itself. Some multiples of 2, 3, and 5. Some multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, 80, etc. Some multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 30, etc. Some multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40, 90, etc. The pupil should notice, on looking over the multiples of 2 and 3, that they have some multiples in common. 6, 12, and 18 are common multiples of 2 and 3. 6 is the smallest common multiple, that is, it is the least common multiple of 2 and 3. The Least Common Multiple (L. C. M.) of two or more numbers is the least number that is a multiple of each of them. WE FINDING THE L. C. M. OF NUMBERS. KNOW 1. That the Least Common Multiple must be larger than any of the numbers. 2. That it must exactly contain each one of the numbers one or more times. Therefore, The Least Common Multiple of two or more numbers must contain every prime factor the greatest number of times it is found in any one of the numbers. ILLUSTRATION: What is the L. C. M. of 12, 18, and 24? EXPLANATION: The prime factors of 12 = 2, 2, 3. The prime factors of 18 = 2, 3, 3. The prime factors of 24 = 2, 2, 2, 3. 2 is found three times in 24. 3 is found twice in 18. L. C. M. of 12, 18, and 24 = 2 × 2 × 2 × 3 × 3, or 72. Principle: The Least Common Multiple of two or more numbers is the product of the prime factors, each prime factor to be taken the greatest number of times it is found in any one number. TO ILLUSTRATE: Find the L. C. M. of 144, 240, 600. WORK: FIRST WAY. 144 = 2 × 2 × 2 × 2 × 3 × 3 240 = 2×2×2×2×3×5 600 = 2 × 2×2×3× 5 × 5 L. C. M. = 2×2×2×2×3×3×5×5, or 3600. NOTE. underlined The series of factors where the prime factors are SECOND WAY. WORK: 2) 144, 240, 600 EXPLANATION: In this method we go through a series of divisions by prime factors that will divide at least two of the numbers. Numbers that cannot be divided should be brought down unchanged. The L. C. M. is the product of the divisors and the final results. FINDING THE L. C. M. OF LARGE NUMBERS. EXAMPLE: Find the L. C. M. of 2862 and 3498. |