## Proposition 23

 If two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number. Let A and B be two numbers relatively prime, and let any number C measure A. I say that C and B are also relatively prime. If C and B are not relatively prime, then some number D measures C and B. Since D measures C, and C measures A, therefore D also measures A. But it also measures B, therefore D measures A and B which are relatively prime, which is impossible. VII.Def.12 Therefore no number measures the numbers C and B. Therefore C and B are relatively prime. Therefore, if two numbers are relatively prime, then any number which measures one of them is relatively prime to the remaining number. Q.E.D.
The proof of this proposition is straightforward.

#### Use of this proposition

This proposition is used in the proof of the next one.

Next proposition: VII.24

Previous: VII.22

 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