---
book: 7
number: 26
id: "VII.26"
kind: "theorem"
uses: ["[[book-7/proposition-24]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.26"
license: "CC-BY-SA-4.0"
---

# VII.26

*If two numbers be prime to two numbers, both to each, their products also will be prime to one another.*

## Proof

For let the two numbers *A*, *B* be prime to the two numbers *C*, *D*; both to each, and let *A* by multiplying *B* make *E*, and let *C* by multiplying *D* make *F*; I say that *E*, *F* are prime to one another.

For, since each of the numbers *A*, *B* is prime to *C*, therefore the product of *A*, *B* will also be prime to *C*. [[book-7/proposition-24|VII. 24]]

But the product of *A*, *B* is *E*; therefore *E*, *C* are prime to one another.

For the same reason *E*, *D* are also prime to one another.

Therefore each of the numbers *C*, *D* is prime to *E*.

Therefore the product of *C*, *D* will also be prime to *E*. [[book-7/proposition-24|VII. 24]]

But the product of *C*, *D* is *F*.

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