Finding Integers Given LCM and GCD

Date: 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 :-)