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If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.

Let A, B be two numbers prime to one another, and let A by multiplying itself make C: I say that B, C are prime to one another.

For let D be made equal to A.

Since A, B are prime to one another, and A is equal to D, therefore D, B are also prime to one another.

Therefore each of the two numbers D, A is prime to B; therefore the product of D, A will also be prime to B. [VII. 24]

But the number which is the product of D, A is C.

Therefore C, B are prime to one another. Q. E. D. 1

1 The Greek, ἐκ τοῦ ἑνὸς αὐτῶν γενόμενος, literally “the number produced from the one of them,” leaves “multiplied into itself” to be understood.

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