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.
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.