---
book: 7
number: 22
id: "VII.22"
kind: "theorem"
uses: ["[[book-7/proposition-17]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.22"
license: "CC-BY-SA-4.0"
---

# VII.22

*The least numbers of those which have the same ratio with them are prime to one another.*

## Proof

Let *A*, *B* be the least numbers of those which have the same ratio with them; I say that *A*, *B* are prime to one another.

For, if they are not prime to one another, some number will measure them.

Let some number measure them, and let it be *C*.

And, as many times as *C* measures *A*, so many units let there be in *D*, and, as many times as *C* measures *B*, so many units let there be in *E*

Since *C* measures *A* according to the units in *D*, therefore *C* by multiplying *D* has made *A*. [[book-7/definitions#Definition 15|VII. Def. 15]]

For the same reason also *C* by multiplying *E* has made *B*.

Thus the number *C* by multiplying the two numbers *D*, *E* has made *A*, *B*; therefore, as *D* is to *E*, so is *A* to *B*; [[book-7/proposition-17|VII. 17]] therefore *D*, *E* are in the same ratio with *A*, *B*, being less than they: which is impossible.

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

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