If an odd number is relatively prime to any number, then it is also relatively prime to double it.

Let the odd number *A* be relatively prime to any number *B,* and let *C* be double of *B.*

I say that *A* is relatively prime to *C.*

If they are not relatively prime, then some number will measure them.

Let a number *D* measure them.

Now *A* is odd, therefore *D* is also odd. And since *D* which is odd measures *C,* and *C* is even, therefore *D* measures the half of *C* also.

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

Therefore *A* cannot but be relatively prime to *C.* Therefore *A* and *C* are relatively prime.

Therefore, *if an odd number is relatively prime to any number, then it is also relatively prime to double it.*

Q.E.D.