Finding Integers Given LCM and GCDDate: 11/05/2002 at 18:37:51 From: Caroline Subject: Finding integers given least common multiple and greatest common divisor My teacher gave the class a "stumper" problem that I can't figure out: The least common multiple of two positive integers is 144. The greatest common divisor is 2. Neither integer is 2. Find both integers. Thanks for your help! Date: 11/06/2002 at 01:33:19 From: Doctor Paul Subject: Re: Finding integers given least common multiple and greatest common divisor gcd(x,y) = 2 144 = lcm(x,y) = x*y/gcd(x,y) = x*y/2 Thus x*y = 288 = 2^5 * 3^2 Notice that if both x and y contained a three in their prime factorization, then gcd(x,y) would be greater than or equal to three. This is too large. So the 3^2 must be entirely in the prime factorization of x or entirely in the prime factorization of y. Neither number is two so we can't have x = 2 and y = 2^4 * 3^2 If we pick x = 2^2, then we have y = 2^3 * 3^2 but in this case, gcd(x,y) = 4. If we pick x = 2^3, then we have y = 2^2 * 3^2 but in this case, gcd(x,y) = 4 If we pick x = 2^4, then we have y = 2 * 3^2 and in this case, gcd(x,y) = 2. So this looks good. x = 16, y = 18 seems to do the trick. I hope this helps. Please write back if you'd like to talk about this some more. - Doctor Paul, The Math Forum http://mathforum.org/dr.math/ Date: 11/07/2002 at 17:21:12 From: Caroline Subject: Thank you (Finding integers given least common multiple and greatest common divisor) Thank you so much - you were a big help :-) |
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