Book 7
29
Let A be a prime number, and let it not measure B; I say that B, A are prime to one another.
29
For, if B, A are not prime to one another, some number will measure them.
29
Let C measure them.
29
Since C measures B, and A does not measure B, therefore C is not the same with A.
29
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.
29
Therefore no number will measure B, A.
29
Therefore A, B are prime to one another. Q. E. D.
PROPOSITION 30.
30
If two numbers by multiplying one another make some number, and any prime number measure the product, it will also measure one of the original numbers.
30
For let the two numbers A, B by multiplying one another make C, and let any prime number D measure C; I say that D measures one of the numbers A, B.
30
For let it not measure A.
30
Now D is prime; therefore A, D are prime to one another. [VII. 29]
30
And, as many times as D measures C, so many units let there be in E.
30
Since then D measures C according to the units in E, therefore D by multiplying E has made C. [VII. Def. 15]
30
Further, A by multiplying B has also made C; therefore the product of D, E is equal to the product of A, B.
30
Therefore, as D is to A, so is B to E. [VII. 19]
