The least common multiple (LCM) of 2 numbers is the smallest number that they both divide evenly into.
For any integer n, show that LCM(9n + 8, 6n + 5) = 54n^2 + 93n + 40.
(In reply to Proof
I don't think that your hand waving actually constitutes an accurate proof. While I don't follow it exactly, it seems that I could follow the same steps to "prove" that (12n - 2) and (8n + 3) are relatively prime, but this not in fact the case for all n. When n = 11, for instance, both are divisible by 13.
Using my method, it is seen that any factor of (12n - 2) and (8n + 3) is also a factor of (12n - 2) - (8n - 3) = (4n - 5). But any factor of these three is also a factor of (8n - 3) - (4n - 5) = (4n + 8). But any factor of these 4 is also a factor of (4n + 8) - (4n - 5) = 13. Therefore, for all n, the GCD of (12n - 2) and (8n - 3) is either 1 or 13.
Edited on November 20, 2013, 1:28 am