6. Reduce 3. 7 and to a common denominator. 11" 3 × 11 × 9=297= first numerator 7× 4× 9=252= second numerator 5× 4×11=220=third numerator 4 × 11 × 9=396=common denominator We multiply the numerator of each fraction by the denominators of the other fractions to find the new numer ators; and we multiply all the denominators together for the new common denominator. 7. Reduce to equivalent fractions with a common denominator: GREATEST COMMON DIVISOR A common divisor of two or more numbers is any number greater than 1 that exactly divides each of them. The greatest common divisor of two or more numbers is the greatest number that exactly divides each of them. Numbers that have no common divisor are said to be prime to one another. When numbers have been resolved into their prime factors, their G. C. D. can be obtained by inspection. 1. Find the G. C. D. of 36, 42, and 84. The G. C. D. required is 6; for each number is divisible by 2 and by 3, and there is no other prime factor common to all three numbers. 3. What is the G. C. D. of 1365 and 1995 ? h. 32, 48, 128. 7. 1326, 3094, 4420. The prime factors of 1365 are 3, 5, 7, 13. The prime factors of 1995 are 3, 5, 7, 19. 3 x 5 x 7 = 105 = G. C. D. of 1365 and 1995 We divide the greater of the two given numbers by the smaller, then the divisor by the remainder; next the last divisor by the new remainder, and so on until there is no remainder. The last divisor will be the G. C. D. required. The G.C.D. of dividend and divisor is also the G. C. D. of divisor and remainder. If the last divisor is unity, the given numbers have no common measure, that is, they are prime to each other. THE G. C. D. OF SEVERAL NUMBERS If the G. C. D. of three numbers be required, we may first find the G. C. D. of two of the numbers. The G. C. D. of this G. C. D. and the third number will be the G. C. D. required. 1. If we require the G. C. D. of 351, 459, and 1017, we first find the G. C. D. of 351 and 459. This G. C. D. is 27. Next we find the G. C. D. of 27 and 1017. This G. C. D. is 9, which is the G. C. D. required. A second method of finding the G. C. D. is to find the prime factors as in finding the L. C. M. 3. What is the G. C. D. of 56, 188, 1200? 256 188 1200 2 28 94 600 As 47 is a prime number, the only common factors are 2 and 2. 4 is the 14 47 300 G. C. D. 4. Find the G. C. D. of 1058, 2196, 10888, 50024, either by extending the first method or by applying the second. 5. Find the G. C. D. of : A 556, 672, 452, 212. C 740, 840, 5060, 3080. B 96, 288, 264, 480. D 1825, 2555, 7300. 6. Find the G. C. D. in 2 by the second method. LEAST COMMON MULTIPLE A multiple of a number is a number exactly divisible by that number. 15, 20, and 45 are multiples of 5. A common multiple of two or more given numbers is any number exactly divisible by each of them. 45 is a common multiple of 3, 5, and 9. The least common multiple of two or more numbers is the least number that is exactly divisible by each of them. 84 is the least common multiple of 3, 7, 14, and 21. A multiple of a number contains all the prime factors of that number. A multiple of two or more numbers contains all the prime factors of those numbers. The least number that exactly contains all the prime factors of two or more numbers must be the least common multiple of those numbers. 1. Find the L. C. M. of 12, 20, 30, 54. 212 20 30 54 2 6 10 15 27 3 3 5 15 27 35 1 5 5 9 1 1 1 9 L. C. M. = 2 × 2 × 3 × 5 × 9 = 540 The L. C. M. of two or more numbers must contain all the different prime factors in the numbers; and if a prime factor is repeated in any of them, it must be repeated in the L. C. M. as many times as it is repeated in that number in which it is most often repeated, and no more. 18: 24 2. Find the L. C. M. of 18, 24, 27, 45. = 2 × 3 × 3 2 × 2 × 2 × 3 27 = 3 × 3 × 3 453 × 3 × 5 |