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:

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

Converse to Corollary

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 hypothesis:

$m \divides n$

By definition of divisibility:

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

Thus:

$q^n = q^{k m} = \paren {q^m}^k$

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Then by Integer Less One divides Power Less One:

$\paren {q^m - 1} \divides \paren {\paren {q^m}^k - 1}$

Hence the result.

$\blacksquare$

Sources

• 1982: P.M. Cohn: Algebra Volume 1 (2nd ed.) ... (previous) ... (next): $\S 2.4$: The rational numbers and some finite fields: Further Exercises $5$