---
book: 9
number: 31
id: "IX.31"
kind: "theorem"
uses: ["[[book-9/proposition-30]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:9.prop.31"
license: "CC-BY-SA-4.0"
---

# IX.31

*If an odd number be prime to any number, it will also be prime to the double of it.*

## Proof

For let the odd number *A* be prime to any number *B*, and let *C* be double of *B*; I say that *A* is prime to *C*.

For, if they are not prime to one another, some number will measure them.

Let a number measure them, and let it be *D*.

Now *A* is odd; therefore *D* is also odd.

And since *D* which is odd measures *C*, and *C* is even, therefore [*D*] will measure the half of *C* also. [[book-9/proposition-30|IX. 30]]

But *B* is half of *C*; therefore *D* measures *B*.

But it also measures *A*; therefore *D* measures *A*, *B* which are prime to one another: which is impossible.

Therefore *A* cannot but be prime to *C*.

Therefore *A*, *C* are prime to one another. Q. E. D.