Kitap 7
21
Therefore A, B are the least of those which have the same ratio with them. Q. E. D.
PROPOSITION 22.
22
The least numbers of those which have the same ratio with them are prime to one another.
22
Let A, B be the least numbers of those which have the same ratio with them; I say that A, B are prime to one another.
22
For, if they are not prime to one another, some number will measure them.
22
Let some number measure them, and let it be C.
22
And, as many times as C measures A, so many units let there be in D, and, as many times as C measures B, so many units let there be in E
22
Since C measures A according to the units in D, therefore C by multiplying D has made A. [VII. Def. 15]
22
For the same reason also C by multiplying E has made B.
22
Thus the number C by multiplying the two numbers D, E has made A, B; therefore, as D is to E, so is A to B; [VII. 17] therefore D, E are in the same ratio with A, B, being less than they: which is impossible.
22
Therefore no number will measure the numbers A, B.
22
Therefore A, B are prime to one another. Q. E. D.
PROPOSITION 23.
23
If two number be prime to one another, the number which measures the one of them will be prime to the remaining number.
23
Let A, B be two numbers prime to one another, and let any number C measure A; I say that C, B are also prime to one another.
23
For, if C, B are not prime to one another, some number will measure C, B.
23
Let a number measure them, and let it be D.
23
Since D measures C, and C measures A, therefore D also measures A.
23
But it also measures B; therefore D measures A, B which are prime to one another: which is impossible. [VII. Def. 12]
23
Therefore no number will measure the numbers C, B.
23
Therefore C, B are prime to one another. Q. E. D.
PROPOSITION 24.
24
If two numbers be prime to any number, their product also will be prime to the same.
24
For let the two numbers A, B be prime to any number C, and let A by multiplying B make D; I say that C, D are prime to one another.
24
For, if C, D are not prime to one another, some number will measure C, D.
