# Integers are Coprime iff Powers are Coprime

## Theorem

Let $a, b \in \Z$ be integers.

Then:

$a \perp b \iff \forall n \in \N: a^n \perp b^n$

That is, two integers are coprime if and only if all their positive integer powers are coprime.

## Proof

The forward implication is shown in Powers of Coprime Numbers are Coprime.

The reverse implication is shown by substituting $n = 1$.

$\blacksquare$