PROPOSITION 30.

If an odd number measure an even number, it will also measure the half of it.

For let the odd number A measure the even number B;

I say that it will also measure the half of it.

A B C
For, since A measures B, let it measure it according to C;

I say that C is not odd.

For, if possible, let it be so.

Then, since A measures B according to C,
therefore A by multiplying C has made B.
Therefore B is made up of odd numbers the multitude of which is odd.
Therefore B is odd: [IX. 23] which is absurd, for by hypothesis it is even.
Therefore C is not odd; therefore C is even.
Thus A measures B an even number of times.
For this reason then it also measures the half of it.
Q. E. D.

Dependency Graph

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Required for

IX.31