---
book: 9
number: 25
id: "IX.25"
kind: "theorem"
uses: ["[[book-9/proposition-24]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.25"
license: "CC-BY-SA-4.0"
---

# IX.25

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

## Proof

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

For let the unit *CD* be subtracted from *BC*; therefore *DB* is even. [[book-7/definitions#Definition 7|VII. Def. 7]]

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

And *CD* is an unit; therefore *CA* is odd. [[book-7/definitions#Definition 7|VII. Def. 7]] Q. E. D.