Two divides Power Plus One iff Odd

Theorem
Let $q, n \in \Z_{>0}$.

Then:
 * $2 \mathrel \backslash \left({q^n + 1}\right)$

$q$ is odd.

In the above, $\backslash$ denotes divisibility.

Proof
By Parity of Integer equals Parity of Positive Power, $q^n$ is even $q$ is even.

Thus it follows that $q^n + 1$ is even $q$ is odd.

The result follows by definition of even integer.