# Power of Two is Even-Times Even Only

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## Theorem

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

Then $a$ is even-times even only.

In the words of Euclid:

*Each of the numbers which are continually doubled beginning from a dyad is even-times even only.*

(*The Elements*: Book $\text{IX}$: Proposition $32$)

## 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.

$\blacksquare$

## Historical Note

This proof is Proposition $32$ of Book $\text{IX}$ of Euclid's *The Elements*.

## Sources

- 1926: Sir Thomas L. Heath:
*Euclid: The Thirteen Books of The Elements: Volume 2*(2nd ed.) ... (previous) ... (next): Book $\text{IX}$. Propositions