|If an odd number is relatively prime to any number, then it is also relatively prime to double it.|
|Let the odd number A be relatively prime to any number B, and let C be double of B.
I say that A is relatively 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.28)
|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 relatively prime to C. Therefore A and C are relatively prime.
|Therefore, if an odd number is relatively prime to any number, then it is also relatively prime to double it.|
Next proposition: IX.32