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