Proposition 23
If two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number.
Let A and B be two numbers relatively prime, and let any number C measure A.
I say that C and B are also relatively prime.
If C and B are not relatively prime, then some number D measures C and B.
Since D measures C, and C measures A, therefore D also measures A. But it also measures B, therefore D measures A and B which are relatively prime, which is impossible.
Therefore no number measures the numbers C and B. Therefore C and B are relatively prime.
Therefore, if two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number.
Q.E.D.