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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.

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

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.

Euclid. Euclid's Elements. Sir Thomas Little Heath. New York. Dover. 1956.

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