# Number whose Half is Odd is Even-Times Odd

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## Theorem

Let $a \in \Z$ be an integer such that $\dfrac a 2$ is an odd integer.

Then $a$ is even-times odd.

In the words of Euclid:

*If a number have its half odd, it is even-times odd only.*

(*The Elements*: Book $\text{IX}$: Proposition $33$)

## Proof

By definition:

- $a = 2 r$

where $r$ is an odd integer.

Thus:

and:

Hence the result by definition of even-times odd integer.

As $r$ is an odd integer it follows that $2 \nmid r$.

Thus $a$ is not divisible by $4$.

Hence $a$ is not even-times even.

$\blacksquare$

## Historical Note

This proof is Proposition $33$ 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