Integer Less One divides Power Less One/Corollary

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

Let:
 * $m \divides n$

where $\divides$ denotes divisibility.

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

Proof
By hypothesis:
 * $m \divides n$

By definition of divisibility:
 * $\exists k \in \Z: k m = n$

Thus:
 * $q^n = q^{k m} = \paren {q^m}^k$

Then by Integer Less One divides Power Less One:
 * $\paren {q^m - 1} \divides \paren {\paren {q^m}^k - 1}$

Hence the result.