I need this for a proof of Wedderburn's theorem: $$q^m - 1 | q^n - 1 \quad \Rightarrow \quad m|n$$ with $q>1 \in \mathbf{N}$ and $m,n \in \mathbf{N}$.
I'd also like to know if it works the other way around: $$m|n \quad \Rightarrow \quad q^m - 1 | q^n - 1.$$
Thanks.