Integer Less One divides Power Less One/Corollary

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Corollary to Integer Less One divides Power Less One

Let $m, n, q \in \Z_{>0}$.

Let:

$m \mathop \backslash n$

where $\backslash$ denotes divisibility.

Then:

$\left({q^m - 1}\right) \mathop \backslash \left({q^n - 1}\right)$


Converse to Corollary

Let $m, n, q \in \Z_{>0}$.

Let

$\left({q^m - 1}\right) \mathop \backslash \left({q^n - 1}\right)$

where $\backslash$ denotes divisibility.

Then:

$m \mathop \backslash n$


Proof

By hypothesis:

$m \mathop \backslash n$

By definition of divisibility:

$\exists k \in \Z: k m = n$

Thus:

$q^n = q^{k m} = \left({q^m}\right)^k$


Then by Integer Less One divides Power Less One:

$\left({q^m - 1}\right) \mathop \backslash \left({\left({q^m}\right)^k - 1}\right)$

Hence the result.

$\blacksquare$


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