Kitap 7
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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.
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Similarly we can also show that, if D do not measure B, it will measure A.
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Therefore D measures one of the numbers A, B. Q. E. D.
PROPOSITION 31.
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Any composite number is measured by some prime number.
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Let A be a composite number; I say that A is measured by some prime number.
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For, since A is composite, some number will measure it.
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Let a number measure it, and let it be B.
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Now, if B is prime, what was enjoined will have been done.
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But if it is composite, some number will measure it.
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Let a number measure it, and let it be C.
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Then, since C measures B, and B measures A, therefore C also measures A.
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And, if C is prime, what was enjoined will have been done.
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But if it is composite, some number will measure it.
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Thus, if the investigation be continued in this way, some prime number will be found which will measure the number before it, which will also measure A.
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For, if it is not found, an infinite series of numbers will measure the number A, each of which is less than the other: which is impossible in numbers.
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Therefore some prime number will be found which will measure the one before it, which will also measure A.
