analysis - Finding that a statement is divisible by a greatest $n$ through induction

I have the following statement.

Find the largest natural number $m$ such that $n^3 - n$ is divisible by $m$ for all $n$ in $\mathbb{N}$. Prove your result.

I have deduced that the greatest natural number $m$ would be $6$ and I am pretty sure that I have to prove this statement through induction. T he base case is pretty obvious but it is the inductive step that is giving me trouble.

Please help!