Geometric Sequence in Lowest Terms has Coprime Extremes/Proof 2

Theorem

A geometric sequence of integers in lowest terms has extremes which are coprime.

In the words of Euclid:

If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another.

(The Elements: Book $\text{VIII}$: Proposition $3$)

Proof

Let $P$ be a geometric sequence of natural numbers of length $n$.

Let the common ratio of $P$ be expressed in canonical form as $\dfrac p q$.

From Construction of Geometric Sequence in Lowest Terms:

$P = \paren {q^n, p q^{n - 1}, p^2 q^{n - 2}, \ldots, p^{n - 1} q, p^n}$

By definition of canonical form:

$p \perp q$

It follows from Powers of Coprime Numbers are Coprime that:

$p^n \perp q^n$

Hence the result.

$\blacksquare$