How can I prove that $a^m-1|a^n-1 \Leftrightarrow m|n$? Suppose that $(a,m,n)\in \mathbb{N}^{3}$, as $a>1$
How can I prove that $a^m-1|a^n-1 \Leftrightarrow m|n$?
I proved that $m|n \implies a^m-1|a^n-1$, but I couldn't prove the reciproque.

From different perspective, we can show by group theory.
it is enough to know that roots of $x^k-1$ is a group with $k$ elements.
suppose $x^m-1$ | $x^n-1$ that means that every root of first polynomial is also a root of second polynomial. 
That means a group of $n$ elements include a group of $m$ elements by lagrange theorem; $m$| $n$.
