|The least numbers of those which have the same ratio with them are relatively prime.|
|Let A and B be the least numbers of those which have the same ratio with them.
I say that A and B are relatively prime.
|If they are not relatively prime, then some number C measures them.
Let there be as many units in D as the times that C measures A, and as many units in E as the times that C measures B.
|Since C measures A according to the units in D, therefore C multiplied by D makes A. For the same reason C multiplied by E makes B.||VII.Def.15|
|Thus the number C multiplied by the two numbers D and E makes A and B, therefore D is to E as A is to B.||VII.17|
|Therefore D and E are in the same ratio with A and B, being less than they, which is impossible. Therefore no number measures the numbers A and B.
Therefore A and B are relatively prime.
|Therefore, the least numbers of those which have the same ratio with them are relatively prime.|
Next proposition: VII.23