Any composite number is measured by some prime number.
任何合数都能被某个素数度量。
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Let A be a composite number; I say that A is measured by some prime number. For, since A is composite, some number will measure it. Let a number measure it, and let it be B.
设A为合数，则存在某数B度量A。
Now, if B is prime, what was enjoined will have been done. But if it is composite, some number will measure it. Let a number measure it, and let it be C.
若B为素数，则结论成立。
Then, since C measures B, and B measures A, therefore C also measures A. And, if C is prime, what was enjoined will have been done. But if it is composite, some number will measure it.
若B为合数，则存在某数C度量B，从而C也度量A。
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. 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. Therefore some prime number will be found which will measure the one before it, which will also measure A.
如此继续，必得素数度量前数，进而度量A；否则将有无穷递减数列，不可能。