Let n $\displaystyle \geq$ 2 and k be any positive integer. Prove that $\displaystyle (n-1)^{2} \mid n^{k}-1$ if and only if $\displaystyle (n-1) \mid k$. I've tried considering $\displaystyle (n-1) \mid (n^{k}-1)/(n-1)$ but it still got me no where.

I'm looking more for a hint than an actual answer as I do want to figure this out mostly on my own.