---
book: 7
number: 29
id: "VII.29"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.29"
license: "CC-BY-SA-4.0"
---

# VII.29

*Any prime number is prime to any number which it does not measure.*

## Proof

Let *A* be a prime number, and let it not measure *B*; I say that *B*, *A* are prime to one another.

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

Let *C* measure them.

Since *C* measures *B*, and *A* does not measure *B*, therefore *C* is not the same with *A*.

Now, since *C* measures *B*, *A*, therefore it also measures *A* which is prime, though it is not the same with it: which is impossible.

Therefore no number will measure *B*, *A*.

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