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 $n = 1$.

$\blacksquare$