## Proposition 27

 If two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime. Let A and B be two relatively prime numbers, let A multiplied by itself make C, and multiplied by C make D, and let B multiplied by itself make E, and multiplied by E make F. I say that C and E are relatively prime, and that D and F are relatively prime. Since A and B are relatively prime, and A multiplied by itself makes C, therefore C and B are relatively prime. VII.25 Since, then, C and B are relatively prime, and B multiplied by itself makes E, therefore C and E are relatively prime. Again, since A and B are relatively prime, and B multiplied by itself makes E, therefore A and E are relatively prime. Since, then, the two numbers A and C are relatively prime to the two numbers B and E, both to each, therefore the product of A and C is relatively prime to the product of B and E. And the product of A and C is D, and the product of B and E is F. VII.26 Therefore D and F are relatively prime. Therefore, if two numbers are relatively prime, and each multiplied by itself makes a certain number, then the products are relatively prime; and, if the original numbers multiplied by the products make certain numbers, then the latter are also relatively prime. Q.E.D.
The proposition states that if two numbers are relatively prime, then their powers are also relatively prime. Explicitly, it only says that their squares are relatively prime, and their cubes are relatively prime, but the way it is used in VIII.2, any powers need to be relatively prime.

The proof of this proposition uses the last two propositions. Assume that a and b are relatively prime. Then applying VII.25 twice, we first get a2 and b relatively prime, then we get a2 and b2 relatively prime.

Again, by VII.25, a and b2 are relatively prime. Now, a is relatively prime to b2, and b is relatively prime to a2, so by VII.26, a3 is relatively prime to b3.

Likewise, higher powers of a and b can be shown to be relatively prime.

#### Use of this proposition

This proposition is used in VIII.2 and VIII.3.

Next proposition: VII.28

Previous: VII.26

 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