Divisibility of Elements in Geometric Sequence of Integers

Theorem
Let $Q_n = \sequence {a_j}_{0 \mathop \le j \mathop \le n}$ be a geometric progression of integers.

Let $j \ne k$.

Then:
 * $\paren {\exists j \in \set {0, 1, \ldots, n - 1}: a_j \divides a_{j + 1} } \iff \paren {\forall j, k \in \set {0, 1, \ldots, n}, j < k: a_j \divides a_k}$

where $\divides$ denotes integer divisibility.

That is:
 * One term of a geometric progression of integers is the divisor of the next term


 * All terms are divisors of all later terms.

Proof
Let $a_j \divides a_{j + 1}$ for some $j \in \set {0, 1, \ldots, n - 1}$.

Then by definition of integer divisibility:
 * $\exists r \in \Z: r a_j = a_{j + 1}$

Thus the common ratio of $Q_n$ is $r$.

So by definition of geometric progression:
 * $\forall j, k \in \set {0, 1, \ldots, n}, j < k: r^{k - j} a_j = a_k$

and so $a_j \divides a_k$.

The converse is trivial.