If two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number.

Let *A* and *B* be two numbers relatively prime, and let any number *C* measure *A.*

I say that *C* and *B* are also relatively prime.

If *C* and *B* are not relatively prime, then some number *D* measures *C* and *B.*

Since *D* measures *C,* and *C* measures *A,* therefore *D* also measures *A.* But it also measures *B,* therefore *D* measures *A* and *B* which are relatively prime, which is impossible.

Therefore no number measures the numbers *C* and *B.* Therefore *C* and *B* are relatively prime.

Therefore, *if two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number.*

Q.E.D.