For all natural numbers n, prove whether the following is true or false:
n3-n is always divisible by 6.
From SQA Advanced Higher Mathematics 2006 Exam Paper
I can choose from the following types of proof:
Proof by contradiction
Proof by contrapositive
Proof by induction
The Attempt at a Solution
I know the statement is true, but proving it has been more difficult than I thought it would be!
I tried proof by induction, but got stuck with trying to prove true for n=k+1. I then tried proving the statement true for n=2k (even number) and n=2m+1 (odd number), but again, I didn't seem to be getting anywhere.
Am I along the right lines, or should I be trying something different?