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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. | 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.

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This proposition is used in the proof of the next one.

Book VII Introduction - Proposition VII.22 - Proposition VII.24.

© 1996
D.E.Joyce
Clark University