Prove that $n(n+1)(n+2)$ is divisible by $6$ for all integers.
I'm not sure if I'm suppose to use division into cases or not. Our teacher ran out of time to go over this in class, and this is on my homework.
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I'd recommend trying this simpler question first:
Prove that $n(n+1)$ is divisible by $2$ (i.e., is even) for all integers $n$.
Note that if either $n$ or $n+1$ is even, then $n(n+1)$ will be even. Must either $n$ or $n+1$ be even?
Once you've solved this problem, the generalization to your original problem should be clear.