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Proposition 29

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.
Q.E.D.

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This proposition is used in the next one and in propositions IX.12 and IX.36.

Book VII Introduction - Proposition VII.28 - Proposition VII.30.

© 1996
D.E.Joyce
Clark University