Kitap 7
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]
30
But D, A are prime to one another, primes are also least, [VII. 21] and the least measure the numbers which have the same ratio the same number of times, the greater the greater and the less the less, that is, the antecedent the antecedent and the consequent the consequent; [VII. 20] therefore D measures B.
30
Similarly we can also show that, if D do not measure B, it will measure A.
30
Therefore D measures one of the numbers A, B. Q. E. D.
PROPOSITION 31.
31
Any composite number is measured by some prime number.
31
Let A be a composite number; I say that A is measured by some prime number.
31
For, since A is composite, some number will measure it.
31
Let a number measure it, and let it be B.
31
Now, if B is prime, what was enjoined will have been done.
31
But if it is composite, some number will measure it.
31
Let a number measure it, and let it be C.
31
Then, since C measures B, and B measures A, therefore C also measures A.
31
And, if C is prime, what was enjoined will have been done.
31
But if it is composite, some number will measure it.
