Book 7
24
Therefore no number will measure the numbers C, D.
24
Therefore C, D are prime to one another. Q. E. D. ὁ ἐξ αὐτῶν γενόμενος, literally the (number) produced from them, will henceforth be translated as their product.
PROPOSITION 25.
25
If two numbers be prime to one another, the product of one of them into itself will be prime to the remaining one.
25
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.
25
For let D be made equal to A.
25
Since A, B are prime to one another, and A is equal to D, therefore D, B are also prime to one another.
25
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]
25
But the number which is the product of D, A is C.
25
Therefore C, B are prime to one another. Q. E. D. The Greek, ὁ ἐκ τοῦ ἑνὸς αὐτῶν γενόμενος, literally the number produced from the one of them, leaves multiplied into itself to be understood.
PROPOSITION 26.
26
If two numbers be prime to two numbers, both to each, their products also will be prime to one another.
26
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.
26
For, since each of the numbers A, B is prime to C, therefore the product of A, B will also be prime to C. [VII. 24]
26
But the product of A, B is E; therefore E, C are prime to one another.
