If an odd number by multiplying an even number make some number, the product will be even.

For let the odd number A by multiplying the even number B make C; I say that C is even.

For, since A by multiplying B has made C, therefore C is made up of as many numbers equal to B as there are units in A. [VII. Def. 15]

And B is even; therefore C is made up of even numbers.

But, if as many even numbers as we please be added together, the whole is even. [IX. 21]

Therefore C is even. Q. E. D.