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Proposition 26

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.
| 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.

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This proposition is used in the proof of the next one.

Book VII Introduction - Proposition VII.25 - Proposition VII.27.

© 1996
D.E.Joyce
Clark University