---
book: 9
number: 27
id: "IX.27"
kind: "theorem"
uses: ["[[book-9/proposition-24]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.27"
license: "CC-BY-SA-4.0"
---

# IX.27

*If from an odd number an even number be subtracted, the remainder will be odd.*

## Proof

For from the odd number *AB* let the even number *BC* be subtracted; I say that the remainder *CA* is odd.

Let the unit *AD* be subtracted; therefore *DB* is even. [[book-7/definitions#Definition 7|VII. Def. 7]]

But *BC* is also even; therefore the remainder *CD* is even. [[book-9/proposition-24|IX. 24]]

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