|Any prime number is relatively prime to any number which it does not measure.|
|Let A be a prime number, and let it not measure B.
I say that B and A are relatively prime.
|If B and A are not relatively prime, then some number C measures them.
Since C measures B, and A does not measure B, therefore C is not the same as A.
|Now, since C measures B and A, therefore it also measures A which is prime, though it is not the same as it, which is impossible. Therefore no number measures B and A.
Therefore A and B are relatively prime.
|Therefore, any prime number is relatively prime to any number which it does not measure.|
Next proposition: VII.30