If two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime.

Let two relatively prime numbers *AB* and *BC* be added.

I say that their sum *AC* is also relatively prime to each of the numbers *AB* and *BC.*

If *CA* and *AB* are not relatively prime, then some number *D* measures *CA* and *AB.*

Since then *D* measures *CA* and *AB,* therefore it also measures the remainder *BC.* But it also measures *BA,* therefore *D* measures *AB* and *BC* which are relatively prime, which is impossible.

Therefore no number measures the numbers *CA* and *AB.* Therefore *CA* and *AB* are relatively prime. For the same reason *AC* and *CB* are also relatively prime. Therefore *CA* is relatively prime to each of the numbers *AB* and *BC.*

Next, let *CA* and *AB* be relatively prime.

I say that *AB* and *BC* are also relatively prime.

If *AB* and *BC* are not relatively prime, then some number *D* measures *AB* and *BC.*

Now, since *D* measures each of the numbers *AB* and *BC,* therefore it also measures the whole *CA.* But it measures *AB,* therefore *D* measures *CA* and *AB* which are relatively prime, which is impossible.

Therefore no number measures the numbers *AB* and *BC.* Therefore *AB* and *BC* are relatively prime.

Therefore, *if two numbers are relatively prime, then their sum is also prime to each of them; and, if the sum of two numbers is relatively prime to either of them, then the original numbers are also relatively prime.*

Q.E.D.

This proposition is used in IX.15.