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