Geometric Sequence with Coprime Extremes is in Lowest Terms

Theorem
Let $G_n = \left\langle{a_0, a_1, \ldots, a_n}\right\rangle$ be a geometric progression of integers.

Let:
 * $a_0 \perp a_n$

where $\perp$ denotes coprimality.

Then $G_n$ is in its lowest terms.