---
book: 7
number: 25
id: "VII.25"
kind: "theorem"
uses: ["[[book-7/proposition-24]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.25"
license: "CC-BY-SA-4.0"
---

# VII.25

*If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.*

## Proof

Let *A*, *B* be two numbers prime to one another, and let *A* by multiplying itself make *C*: I say that *B*, *C* are prime to one another.

For let *D* be made equal to *A*.

Since *A*, *B* are prime to one another, and *A* is equal to *D*, therefore *D*, *B* are also prime to one another.

Therefore each of the two numbers *D*, *A* is prime to *B*; therefore the product of *D*, *A* will also be prime to *B*. [[book-7/proposition-24|VII. 24]]

But the number which is the product of *D*, *A* is *C*.

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