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Proposition 25

If two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one.
Let A and B be two numbers relatively prime, and let A multiplied by itself make C. I say that B and C are relatively prime.
| Make D equal to A.
Since A and B are relatively prime, and A equals D, therefore D and B are also relatively prime. Therefore each of the two numbers D and A is relatively prime to B. Therefore the product of D and A is also relatively prime to B. | VII.24
But the number which is the product of D and A is C. Therefore C and B are relatively prime.
Therefore, if two numbers are relatively prime, then the product of one of them with itself is relatively prime to the remaining one.
Q.E.D.

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This proposition is used in VII.27 and IX.15.

Book VII Introduction - Proposition VII.24 - Proposition VII.26.

© 1996
D.E.Joyce
Clark University