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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.

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

Book VII Introduction - Proposition VII.26 - Proposition VII.28.

© 1996
D.E.Joyce
Clark University