---
book: 9
number: 34
id: "IX.34"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.34"
license: "CC-BY-SA-4.0"
---

# IX.34

*If a number neither be one of those which are continually doubled from a dyad, nor have its half odd, it is both eventimes even and even-times odd*.

## Proof

For let the number *A* neither be one of those doubled from a dyad, nor have its half odd; I say that *A* is both even-times even and even-times odd.

Now that *A* is even-times even is manifest; for it has not its half odd. [[book-7/definitions#Definition 8|VII. Def. 8]]

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

For, if we bisect *A*, then bisect its half, and do this continually, we shall come upon some odd number which will measure *A* according to an even number.

For, if not, we shall come upon a dyad, and *A* will be among those which are doubled from a dyad: which is contrary to the hypothesis.

Thus *A* is even-times odd.

But it was also proved even-times even.

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