Elements of Geometric Sequence from One which are Powers of Number

Theorem
Let $G_n = \sequence {a_n}_{0 \mathop \le i \mathop \le n}$ be a geometric sequence of integers.

Let $a_0 = 1$.

Then:
 * $\forall m, k \in \set {1, \ldots, n}: k \divides m \implies a_m$ is a power of $k$

where $\divides$ denotes divisibility.

Proof
By Form of Geometric Sequence of Integers from One, the general term of $G_n$ can be expressed as:
 * $a_j = q^j$

for some $q \in \Z$.

Let $k, m \in \set {1, 2, \ldots, n}$ such that $k \divides m$.

By definition of divisibility:
 * $\exists r \in \Z: m = r k$

Then:

That is, $a_m$ is a power of $k$.