Book 7
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.
26
For the same reason E, D are also prime to one another.
26
Therefore each of the numbers C, D is prime to E.
26
Therefore the product of C, D will also be prime to E. [VII. 24]
26
But the product of C, D is F.
26
Therefore E, F are prime to one another. Q. E. D.
PROPOSITION 27.
27
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].
27
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.
27
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. [VII. 25]
27
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. [id.]
27
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.]
