Elements of Geometric Sequence from One Divisible by Prime

Theorem
Let $G_n = \left\langle{a_n}\right\rangle_{0 \mathop \le i \mathop \le n}$ be a geometric progression of integers.

Let $a_0 = 1$.

Let $p$ be a prime number such that:
 * $p \mathop \backslash a_n$

where $\backslash$ denotes divisibility.

Then $p \mathop \backslash a_1$.

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

for some $q \in \Z$.

Thus by hypothesis:
 * $p \mathop \backslash q^n$

From Euclid's Lemma for Prime Divisors: General Result:
 * $p \mathop \backslash q$

Hence the result.