IX.27

If from an odd number an even number be subtracted, the remainder will be odd.

Proof

For from the odd number AB let the even number BC be subtracted; I say that the remainder CA is odd.

Let the unit AD be subtracted; therefore DB is even. VII. Def. 7

But BC is also even; therefore the remainder CD is even. IX. 24

Therefore CA is odd. VII. Def. 7 Q. E. D.