## Proposition 34

 If an [even] number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and even-times odd. Let the [even] number A neither be one of those doubled from a dyad, nor have its half odd. I say that A is both even-times even and even-times odd. Now that A is even-times even is manifest, for it has not its half odd. VII.Def.8 I say next that it is also even-times odd. If we bisect A, then bisect its half, and do this continually, we shall come upon some odd number which measures A according to an even number. If not, we shall come upon a dyad, and A will be among those which are doubled from a dyad, which is contrary to the hypothesis. Thus A is even-times odd. But it was also proved even-times even. Therefore A is both even-times even and even-times odd. Therefore, if an [even] number neither is one of those which is continually doubled from a dyad, nor has its half odd, then it is both even-times even and even-times odd. Q.E.D.
This completes the series of propostions on even and odd numbers that started with IX.21.

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