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Proposition 22

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.
Q.E.D.

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This proposition is used in propositions VIII.2, VIII.3, and IX.15.

Book VII Introduction - Proposition VII.21 - Proposition VII.23.

© 1996
D.E.Joyce
Clark University