---
book: 7
number: 23
id: "VII.23"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.23"
license: "CC-BY-SA-4.0"
---

# VII.23

*If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.*

## Proof

Let *A*, *B* be two numbers prime to one another, and let any number *C* measure *A*; I say that *C*, *B* are also prime to one another.

For, if *C*, *B* are not prime to one another, some number will measure *C*, *B*.

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

Since *D* measures *C*, and *C* measures *A*, therefore *D* also measures *A*.

But it also measures *B*; therefore *D* measures *A*, *B* which are prime to one another: which is impossible. [[book-7/definitions#Definition 12|VII. Def. 12]]

Therefore no number will measure the numbers *C*, *B*.

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