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.

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I say that B and C are relatively prime.

Make D equal to A.

VII.24

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.

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.

Guide

 This proposition says that if a is relatively prime to b, then a2 is also relatively prime to b.

It’s a special case of the previous proposition and hardly needs its own enunciation. It is used in VII.27 and IX.15.