---
book: 7
number: 28
id: "VII.28"
kind: "theorem"
uses: []
source: "https://scaife.perseus.org/reader/urn:cts:greekLit:tlg1799.tlg001.perseus-eng2:7.prop.28"
license: "CC-BY-SA-4.0"
---

# VII.28

*If two numbers be prime to one another, the sum will also be prime to each of them; and, if the sum of two numbers be prime to any one of them, the original numbers will also be prime to one another.*

## Proof

For let two numbers *AB*, *BC* prime to one another be added; I say that the sum *AC* is also prime to each of the numbers *AB*, *BC*.

For, if *CA*, *AB* are not prime to one another, some number will measure *CA*, *AB*.

Let a number measure them, and let it be *D*.

Since then *D* measures *CA*, *AB*, therefore it will also measure the remainder *BC*.

But it also measures *BA*; therefore *D* measures *AB*, *BC* which are prime to one another: which is impossible. [[book-7/definitions#Definition 12|VII. Def. 12]]

Therefore no number will measure the numbers *CA*, *AB*; therefore *CA*, *AB* are prime to one another.

For the same reason *AC*, *CB* are also prime to one another.

Therefore *CA* is prime to each of the numbers *AB*, *BC*.

Again, let *CA*, *AB* be prime to one another; I say that *AB*, *BC* are also prime to one another.

For, if *AB*, *BC* are not prime to one another, some number will measure *AB*, *BC*.

Let a number measure them, and let it be *D*.

Now, since *D* measures each of the numbers *AB*, *BC*, it will also measure the whole *CA*.

But it also measures *AB*; therefore *D* measures *CA*, *AB* which are prime to one another: which is impossible. [[book-7/definitions#Definition 12|VII. Def. 12]]

Therefore no number will measure the numbers *AB*, *BC*.

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