---
book: 9
number: 32
id: "IX.32"
kind: "theorem"
uses: ["[[book-9/proposition-13]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.32"
license: "CC-BY-SA-4.0"
---

# IX.32

*Each of the numbers which are continually doubled beginning from a dyad is even-times even only.*

## Proof

For let as many numbers as we please, *B*, *C*, *D*, have been continually doubled beginning from the dyad *A*; I say that *B*, *C*, *D* are eventimes even only.

Now that each of the numbers *B*, *C*, *D* is even-times even is manifest; for it is doubled from a dyad.

I say that it is also even-times even only.

For let an unit be set out.

Since then as many numbers as we please beginning from an unit are in continued proportion, and the number *A* after the unit is prime, therefore *D*, the greatest of the numbers *A*, *B*, *C*, *D*, will not be measured by any other number except *A*, *B*, *C*. [[book-9/proposition-13|IX. 13]]

And each of the numbers *A*, *B*, *C* is even; therefore *D* is even-times even only. [[book-7/definitions#Definition 8|VII. Def. 8]]

Similarly we can prove that each of the numbers *B*, *C* is even-times even only. Q. E. D.