Powers of Coprime Numbers are Coprime

Theorem
Let $a, b$ be coprime integers:
 * $a \perp b$

Then:
 * $\forall n \in \N_{>0}: a^n \perp b^n$

Proof
Proof by induction:

Let $a \perp b$.

For all $n \in \N_{> 0}$, let $\map P n$ be the proposition:
 * $a^n \perp b^n$

$\map P 1$ is true, as this just says:
 * $a \perp b$

Basis for the Induction
By :
 * $a^2 \perp b$

Again, by :
 * $a^2 \perp b^2$

This is our basis for the induction.

Induction Hypothesis
Now we need to show that, if $\map P k$ is true, where $k \ge 2$, then it logically follows that $\map P {k + 1}$ is true.

So this is our induction hypothesis:
 * $a^k \perp b^k$

Then we need to show:
 * $a^{k + 1} \perp b^{k + 1}$

Induction Step
This is our induction step:

So $\map P k \implies \map P {k + 1}$ and the result follows by the Principle of Mathematical Induction.

Therefore:
 * $\forall n \in \N_{>0}: a^n \perp b^n$