---
book: 8
number: 3
id: "VIII.3"
kind: "theorem"
uses: ["[[book-7/proposition-33]]", "[[book-8/proposition-2]]", "[[book-7/proposition-22]]", "[[book-7/proposition-27]]"]
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:8.prop.3"
license: "CC-BY-SA-4.0"
---

# VIII.3

*If as many numbers as we please in continued proportion be the least of those which have the same ratio with them, the extremes of them are prime to one another*.

## Proof

Let as many numbers as we please, *A*, *B*, *C*, *D*, in continued proportion be the least of those which have the same ratio with them; I say that the extremes of them *A*, *D* are prime to one another.

For let two numbers *E*, *F*, the least that are in the ratio of *A*, *B*, *C*, *D*, be taken, [[book-7/proposition-33|VII. 33]] then three others *G*, *H*, *K* with the same property; and others, more by one continually, [[book-8/proposition-2|VIII. 2]] until the multitude taken becomes equal to the multitude of the numbers *A*, *B*, *C*, *D*.

Let them be taken, and let them be *L*, *M*, *N*, *O*.

Now, since *E*, *F* are the least of those which have the same ratio with them, they are prime to one another. [[book-7/proposition-22|VII. 22]]

And, since the numbers *E*, *F* by multiplying themselves respectively have made the numbers *G*, *K*, and by multiplying the numbers *G*, *K* respectively have made the numbers *L*, *O*, [[book-8/proposition-2|VIII. 2, Por.]] therefore both *G*, *K* and *L*, *O* are prime to one another. [[book-7/proposition-27|VII. 27]]

And, since *A*, *B*, *C*, *D* are the least of those which have the same ratio with them, while *L*, *M*, *N*, *O* are the least that are in the same ratio with *A*, *B*, *C*, *D*, and the multitude of the numbers *A*, *B*, *C*, *D* is equal to the multitude of the numbers *L*, *M*, *N*, *O*, therefore the numbers *A*, *B*, *C*, *D* are equal to the numbers *L*, *M*, *N*, *O* respectively; therefore *A* is equal to *L*, and *D* to *O*.

And *L*, *O* are prime to one another.

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