If two numbers be prime to two numbers, both to each, their products also will be prime to one another.
若两数分别与两数互质，则它们的乘积也互质。
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For let the two numbers A, B be prime to the two numbers C, D; both to each, and let A by multiplying B make E, and let C by multiplying D make F; I say that E, F are prime to one another. For, since each of the numbers A, B is prime to C, therefore the product of A, B will also be prime to C.
设A、B与C、D分别互质，且A乘B得E，C乘D得F。
[VII. 24] But the product of A, B is E; therefore E, C are prime to one another. For the same reason E, D are also prime to one another.
因A、B均与C互质，由VII.24知A、B之积E与C互质。
Therefore each of the numbers C, D is prime to E. Therefore the product of C, D will also be prime to E.
同理，E与D也互质，故C、D均与E互质。
[VII. 24] But the product of C, D is F.
由VII.24，C、D之积F与E互质，即E、F互质。