内容 第7卷 · 204
命题 Propositio VII.29
Any prime number is prime to any number which it does not measure.
如果一个素数不度量另一个数，那么这两个数互素。
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分步证明Step-by-step proof
1 / 4Let A be a prime number, and let it not measure B; I say that B, A are prime to one another. For, if B, A are not prime to one another, some number will measure them.
设A是素数，且A不度量B。假设B和A不互素，则存在某个数C度量它们。
Let C measure them.
由于C度量B，而A不度量B，因此C与A不同。
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.
因为C度量B和A，所以C也度量素数A，但C与A不同，这是不可能的。
Therefore no number will measure B, A.
因此，没有数能度量B和A，即B和A互素。
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