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 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

© 1996, 2002
D.E.Joyce
Clark University