If two numbers be prime to any number, their product also will be prime to the same.
若两数均与某数互质，则它们的乘积也与该数互质。
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For let the two numbers A, B be prime to any number C, and let A by multiplying B make D; I say that C, D are prime to one another. For, if C, D are not prime to one another, some number will measure C, D. Let a number measure them, and let it be E.
设两数A、B均与C互质，且A乘B得D。假设C与D不互质，则存在某数E同时量尽C和D。
Now, since C, A are prime to one another, and a certain number E measures C, therefore A, E are prime to one another. [VII. 23] As many times, then, as E measures D, so many units let there be in F; therefore F also measures D according to the units in E. [VII. 16] Therefore E by multiplying F has made D.
由于C与A互质，且E量尽C，故A与E互质（据VII.23）。设E量尽D的次数为F，则E乘F得D（据VII.16及定义15）。
[VII. Def. 15] But, further, A by multiplying B has also made D; therefore the product of E, F is equal to the product of A, B. But, if the product of the extremes be equal to that of the means, the four numbers are proportional; [VII. 19] therefore, as E is to A, so is B to F. But A, E are prime to one another, numbers which are prime to one another are also the least of those which have the same ratio, [VII. 21] and the least numbers of those which have the same ratio with them measure those 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 E measures B.
又A乘B得D，故E乘F等于A乘B。由比例性质（VII.19），得E比A等于B比F。因A与E互质，它们是最小比数（VII.21），故E量尽B（VII.20）。
But it also measures C; therefore E measures B, C which are prime to one another: which is impossible. [VII. Def. 12] Therefore no number will measure the numbers C, D.
但E也量尽C，故E量尽互质的B与C，矛盾（定义12）。因此没有数能同时量尽C与D，即C与D互质。