| 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. | VII.Def.12 | |
| 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. | ||
Book VII Introduction - Proposition VII.22 - Proposition VII.24.