---
book: 9
number: 33
id: "IX.33"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.33"
license: "CC-BY-SA-4.0"
---

# IX.33

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

## Proof

For let the number *A* have its half odd; I say that *A* is even-times odd only.

Now that it is even-times odd is manifest; for the half of it, being odd, measures it an even number of times. [[book-7/definitions#Definition 9|VII. Def. 9]]

I say next that it is also even-times odd only.

For, if *A* is even-times even also, it will be measured by an even number according to an even number; [[book-7/definitions#Definition 8|VII. Def. 8]] so that the half of it will also be measured by an even number though it is odd: which is absurd.

Therefore *A* is even-times odd only. Q. E. D.