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Proposition 31

If an odd number is prime to any number, then it is also prime to double it.
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.
| If they are not relatively prime, then some number will measure them. Let a number D measure them.
Now A is odd, therefore D is also odd. And since D which is odd measures C, and C is even, therefore D measures 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 and B which are relatively prime, which is impossible. Therefore A cannot but be prime to C. Therefore A and C are relatively prime.
Therefore, if an odd number is prime to any number, then it is also prime to double it.
Q.E.D.

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(Forthcoming)

Book IX Introduction - Proposition IX.30 - Proposition IX.32.

© 1996
D.E.Joyce
Clark University