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.*

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.*

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.*

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.*

Q.E.D.