| 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. |
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| 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. |
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| 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. | ||
| Q.E.D. | ||
Book VII Introduction - Proposition VII.21 - Proposition VII.23.