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