## Proposition 25

 If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one. Let A and B be two numbers relatively prime, and let A multiplied by itself make C. I say that B and C are relatively prime. Make D equal to A. Since A and B are relatively prime, and A equals D, therefore D and B are also relatively prime. Therefore each of the two numbers D and A is relatively prime to B. Therefore the product of D and A is also relatively prime to B. VII.24 But the number which is the product of D and A is C. Therefore C and B are relatively prime. Therefore, if two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one. Q.E.D.
This is a special case of the previous proposition. It is used in VII.27 and IX.15.

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 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