PROPOSITION 29.

Any prime number is 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, A are prime to one another.

A B C
For, if B, A are not prime to one another, some number will measure them.

Let C measure them.

Since C measures B, and A does not measure B,
therefore C is not the same with A.
Now, since C measures B, A,
therefore it also measures A which is prime, though it is not the same with it : which is impossible.
Therefore no number will measure B, A.
Therefore A, B are prime to one another.
Q. E. D.

Dependency Graph

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Required for

VII.30, IX.12, IX.13, IX.14, IX.32, IX.36

See Commentary on Proposition VII.29.