Power of Two is Even-Times Even Only

Theorem
Let $a > 2$ be a power of $2$.

Then $a$ is even-times even only.

Proof
As $a$ is a power of $2$ greater than $2$:
 * $\exists k \in \Z_{>1}: a = 2^k$

Thus:
 * $a = 2^2 2^{k - 2}$

and so has $2^2 = 4$ as a divisor.

Let $b$ be an odd number.

By definition:
 * $b \nmid 2$

The result follows by Integer Coprime to Factors is Coprime to Whole.