If two numbers be prime to one another, and each by multiplying itself make a certain number, the products will be prime to one another; and, if the original numbers by multiplying the products make certain numbers, the latter will also be prime to one another [and this is always the case with the extremes].
若两数互质，则每个数自乘所得之积互质；且原数乘此积所得之数亦互质（此性质恒适用于两端）。
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Let A, B be two numbers prime to one another, let A by multiplying itself make C, and by multiplying C make D, and let B by multiplying itself make E, and by multiplying E make F; I say that both C, E and D, F are prime to one another. For, since A, B are prime to one another, and A by multiplying itself has made C, therefore C, B are prime to one another.
设A、B互质，A自乘得C，A乘C得D；B自乘得E，B乘E得F。
[VII. 25] Since then C, B are prime to one another, and B by multiplying itself has made E, therefore C, E are prime to one another.
因A、B互质，且A自乘得C，故C与B互质（VII.25）。
[id.] Again, since A, B are prime to one another, and B by multiplying itself has made E, therefore A, E are prime to one another. [id.] Since then the two numbers A, C are prime to the two numbers B, E, both to each, therefore also the product of A, C is prime to the product of B, E.
因C与B互质，且B自乘得E，故C与E互质（VII.25）。同理，A与E互质。
[VII. 26] And the product of A, C is D, and the product of B, E is F.
因A、C与B、E分别互质，故A、C之积D与B、E之积F互质（VII.26）。