Number whose Half is Odd is Even-Times Odd

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:

$a$ has an even divisor

and:

$a$ has an odd divisor.

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