Integer Less One divides Power Less One/Corollary/Converse

Converse of Corollary to Integer Less One divides Power Less One
Let $m, n, q \in \Z_{>0}$.

Let
 * $\paren {q^m - 1} \divides \paren {q^n - 1}$

where $\divides$ denotes divisibility.

Then:
 * $m \divides n$

Proof
By the Division Theorem:
 * $\exists a, b \in \Z: n = m a + b, 0 \le b < m$

Then:
 * $q^n \equiv q^{m a} q^b \equiv q^b \equiv 1 \pmod {\paren {q^m - 1} }$

But:
 * $q^b - 1 < q^m - 1$

So:
 * $q^b - 1 = 0$

and so $b = 0$.

Hence the result.