The least common multiple (LCM) of 2 numbers is the smallest number that they both divide evenly into.
e.g.:
LCM(8,10)=40
LCM(17,11)=187

For any integer n, show that LCM(9n + 8, 6n + 5) = 54n^2 + 93n + 40.

The formula on the right is for the product (9n+8)(6n+5), which us equal to the left hand side if and only if 9n+8 and 6n+5 are relatively prime.

6n+5 is definitely odd, so the two numbers do not share a factor of 2. Neither is a multiple of 3. For 6n+5 to be a multiple of 5, n has to be a multiple of 5, in which case the 9n+8 is not a multiple of 5.

The two numbers differ by 3n+3, which is divisible by 3, which is useless for getting a common factor of the two numbers as we've seen the numbers themselves are not divisible by 3. They differ by a multiple of n as well. Could some factor of n be a factor of both?

If a proposed common factor of both is a factor of n, then it must divide both 8 and 5, so there is no such common factor, and indeed the LCM is just the product, and therefore equal to the RHS.

 

Posted by Charlie on 2013-11-19 18:11:31