234. The factors that are common to two or more numbers may be taken out of the numbers simultaneously, as follows: By the tests given in ? 222, 4 (22) is seen to be common to the numbers, and 9 (32) common to the resulting quotients; the quotients 3, 11, 40, have no common factor, therefore 22 and 3o are the only factors common to the numbers. Hence, their G. C. M. is 22 × 32, or 36. 235. When it is required to find the G. C. M. of two or more numbers which cannot readily be resolved into factors, the method to be employed is as follows: This method depends upon two principles: 1. That every factor of a number is also a factor of every multiple of that number. 24, 36, etc. Thus 4, which is a factor of 12, is a factor also of 2. That any common factor of two numbers is also a factor of their sum and of their difference. Thus 4, which is a common factor of 24 and 36, is also a factor of 60 and 12. Apply these principles to this example : Since 6 is a factor of itself and of 12, it is, by (2), a factor of 18. Since 6 is a factor of 18, it is, by (1), a factor of 2 x 18, or 36; and therefore, by (2), it is a factor of 36 + 12, or 48. Hence, 6 is a common factor of 18 and 48. Again, every common factor of 18 and 48 is, by (1), a factor of 2× 18, or 36; and, by (2), a factor of 48 – 36, or 12. Every such factor, being now a common factor of 18 and 12 is, by (2), a factor of 1812, or 6. Therefore, the greatest common factor of 18 and 48 is contained in 6, and cannot be greater than 6. And 6, which has been shown to be a common factor of 18 and 48, must be their G. C. M. 236. It will be seen that every remainder in the course of the operation contains, as a factor of itself, the G. C. M. sought; and that this is the greatest factor common to that remainder and the preceding divisor. the Therefore, a factor which is discovered, at any stage of process, to belong to one of the numbers and not to the other, may be ejected; and a factor which is discovered, at any stage of the process, to belong to both numbers, may be taken out and reserved as a factor of the G. C. M. (1) Find the G. C. M. of 11,237 and 12,559. 11237)12559(1 11237 21322 661)11237(17 661 4627 4627 Hence, the G. C. M. is 661. The factor 2 is thrown out of the first remainder 1322, for it is not contained in 11,237, and therefore is not a factor of the G. C. M. sought. (2) Find the G. C. M. of 269,178 and 352,002. 6269178 352002 44863) 58667(1 44863 4)13804 3451) 44863(13 3451 10353 10353 Hence, the G. C. M. is 6 x 3451. The common factor 6 is first taken out from both numbers. From the remainder 13,804 the factor 4, which is prime to 44,863, is ejected. The resulting number 3451 is contained exactly in 44,863, and therefore the G. C. M. is 6 x 3451. 237. To find the G. C. M. of several large numbers: Find the G. C. M. of two of the numbers; then of that result and a third number; then of that result and a fourth; and so on. The last G. C. M. is the one required. The work can generally be very much shortened by removing from each of the numbers all factors less than 13. Of these, the factors common to the numbers must be retained as factors of the G. C. M. Find the G. C. M. of 3555, 4977, and 6636. Hence, the G. C. M. is 3 x 79, or 237. The common factor 3 is first taken out and reserved as a factor of the G. C. M. From the resulting quotients the factors less than 13 are removed, and 79 is found to be common to the numbers. Hence, their G. C. M. is 3 x 79. EXERCISE XV. Find the G. C. M. of 1. 855, 1,197; 1,596. etc. 5. 1,177, 1,391, 1,819. 6. 4,939, 1,347, 3,143. 7. 740, 333, 296. 8. 833, 1,785, 1,309. 9. 7,326, 8,547, 9,768, 22,755. 10. 4,994, 7,491, 9,988, 12,485, 16,571. LEAST COMMON MULTIPLE. 238. The multiples of 2 are 2, 4, 6, 8, 10, 12, 14, 16, 18, The multiples of 3 are 3, 6, 9, 12, 15, 18, etc. The multiples common to 2 and 3 are seen to be 6, 12, 18, etc.; and of these, 6 is the least. 239. The multiples that two or more numbers have in common are called their common multiples, and the least of these is called their Least Common Multiple, which is indicated by the letters L. C. M. Find the L. C. M. of 7, 8, 9, 10. The L. C. M. of 7, 8, 9, 10 must contain the factor 7, else it would not be a multiple of 7. It must also contain 23 to be a multiple of 8, and 32 to be a multiple of 9. contain the factors 2 and 5 to be a multiple of 10. It must That is, the L. C. M. of 7, 8, 9, 10, must contain the factors 7, 23, 32, and 5; therefore it is 7 × 23 × 32 × 5, or 2520. Hence, 240. To find the L. C. M. of two or more numbers: Separate each number into its prime factors. Find the product of these powers. |