---
book: 7
number: 27
id: "VII.27"
kind: "theorem"
uses: ["[[book-7/proposition-25]]", "[[book-7/proposition-26]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.27"
license: "CC-BY-SA-4.0"
---

# VII.27

*If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another* [*and this is always the case with the extremes*].

## Proof

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

For, since *A*, *B* are prime to one another, and *A* by multiplying itself has made *C*, therefore *C*, *B* are prime to one another. [[book-7/proposition-25|VII. 25]]

Since then *C*, *B* are prime to one another, and *B* by multiplying itself has made *E*, therefore *C*, *E* are prime to one another. [*id*.]

Again, since *A*, *B* are prime to one another, and *B* by multiplying itself has made *E*, therefore *A*, *E* are prime to one another. [*id*.]

Since then the two numbers *A*, *C* are prime to the two numbers *B*, *E*, both to each, therefore also the product of *A*, *C* is prime to the product of *B*, *E*. [[book-7/proposition-26|VII. 26]]

And the product of *A*, *C* is *D*, and the product of *B*, *E* is *F*.

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