[Folio 0423 | p. 417]

PROPOSITION 31.

If an odd number be prime to any number, it will also be prime to the double of it.

For let the odd number A be prime to any number B, and let C be double of B;

I say that A is prime to C.

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

[Folio 0424 | p. 418]

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,

[IX. 30]therefore [D] will measure the half of C also.

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.

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