| If two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime. | ||
| Let the two numbers A and B be relatively prime to the two numbers C and D, both to each, and let A multiplied by B make E, and let C multiplied by D make F.
I say that E and F are relatively prime. |
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| Since each of the numbers A and B is relatively prime to C, therefore the product of A and B is also relatively prime to C. But the product of A and B is E, therefore E and C are relatively prime. For the same reason E and D are also relatively prime. Therefore each of the numbers C and D is relatively prime to E. | VII.24 | |
| Therefore the product of C and D is also relatively prime to E. But the product of C and D is F. Therefore E and F are relatively prime. | VII.24 | |
| Therefore, if two numbers are relatively prime to two numbers, both to each, then their products are also relatively prime. | ||
| Q.E.D. | ||
Book VII Introduction - Proposition VII.25 - Proposition VII.27.