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

# IX.22

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

## Proof

For let as many odd numbers as we please, *AB*, *BC*, *CD*, *DE*, even in multitude, be added together; I say that the whole *AE* is even.

For, since each of the numbers *AB*, *BC*, *CD*, *DE* is odd, if an unit be subtracted from each, each of the remainders will be even; [[book-7/definitions#Definition 7|VII. Def. 7]] so that the sum of them will be even. [[book-9/proposition-21|IX. 21]]

But the multitude of the units is also even.

Therefore the whole *AE* is also even. [[book-9/proposition-21|IX. 21]] Q. E. D.