---
book: 9
number: 23
id: "IX.23"
kind: "theorem"
uses: ["[[book-9/proposition-22]]", "[[book-9/proposition-21]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.23"
license: "CC-BY-SA-4.0"
---

# IX.23

*If as many odd numbers as we please be added together, and their multitude be odd, the whole will also be odd.*

## Proof

For let as many odd numbers as we please, *AB*, *BC*, *CD*, the multitude of which is odd, be added together; I say that the whole *AD* is also odd.

Let the unit *DE* be subtracted from *CD*; therefore the remainder *CE* is even. [[book-7/definitions#Definition 7|VII. Def. 7]]

But *CA* is also even; [[book-9/proposition-22|IX. 22]] therefore the whole *AE* is also even. [[book-9/proposition-21|IX. 21]]

And *DE* is an unit.

Therefore *AD* is odd. [[book-7/definitions#Definition 7|VII. Def. 7]] Q. E. D.