For, if they are not prime to one another, some number will measure them.

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

Now A is odd; therefore D is also odd.

And since D which is odd measures C, and C is even, therefore [D] will measure the half of C also. [[IX. 30](elem.9.30)]

But B is half of C; therefore D measures B.

But it also measures A; therefore D measures A, B which are prime to one another: which is impossible.

Therefore A cannot but be prime to C.

Therefore A, C are prime to one another. Q. E. D.