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.
Any common factor of (9n + 8) and (6n + 5) is also a factor of their difference, (3n + 3).
Any common factor of (9n + 8), (6n + 5) and (3n + 3) is also a factor of (6n + 5) - (3n + 3), i.e. (3n + 2).
Any common factor of (9n + 8), (6n + 5), (3n + 3) and (3n + 2) is also a factor of (3n + 3) - (3n + 2), i.e. 1.
But the only factor of 1 is 1, so, all of the terms mentioned are relatively prime, including (9n + 8) and (6n + 5).
Edited on November 20, 2013, 9:11 am
Another Proof
