## 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.
This proposition is the converse of the last one. Together they say that a ratio a:b is reduced to lowest terms if and only if a is relatively prime to b.

#### Use of this proposition

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

Next proposition: VII.23

Previous: VII.21

 Select from Book VII Book VII intro VII.Def.1-2 VII.Def.3-5 VII.Def.6-10 VII.Def.11-14 VII.Def.15-19 VII.Def.20 VII.Def.21 VII.Def.22 VII.1 VII.2 VII.3 VII.4 VII.5 VII.6 VII.7 VII.8 VII.9 VII.10 VII.11 VII.12 VII.13 VII.14 VII.15 VII.16 VII.17 VII.18 VII.19 VII.20 VII.21 VII.22 VII.23 VII.24 VII.25 VII.26 VII.27 VII.28 VII.29 VII.30 VII.31 VII.32 VII.33 VII.34 VII.35 VII.36 VII.37 VII.38 VII.39 Select book Book I Book II Book III Book IV Book V Book VI Book VII Book VIII Book IX Book X Book XI Book XII Book XIII Select topic Introduction Table of Contents Geometry applet About the text Euclid Web references A quick trip