If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one.

Let *A* and *B* be two numbers relatively prime, and let *A* multiplied by itself make *C.*

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

Make *D* equal to *A.*

Since *A* and *B* are relatively prime, and *A* equals *D,* therefore *D* and *B* are also relatively prime. Therefore each of the two numbers *D* and *A* is relatively prime to *B.* Therefore the product of *D* and *A* is also relatively prime to *B.*

But the number which is the product of *D* and *A* is *C.* Therefore *C* and *B* are relatively prime.

Therefore, *if two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one.*

Q.E.D.

It’s a special case of the previous proposition and hardly needs its own enunciation. It is used in VII.27 and IX.15.