Hello gate13 Originally Posted by

**gate13** Good day to all,

I have been asked the following question:

**Prove that if p is prime and p|n^2 then p|n**

I have attached my proof and was hoping to get an opinion on it, to see if I have understood the material and where I need to tighten up the proof.

Thanks

I'm no expert on Number Theory, but, since no-one else has offered any comment, your proof looks basically sound to me. I think you could say, after the line, '$\displaystyle p \in$ prime decomposition of $\displaystyle n^2$'

Assume without loss of generality that $\displaystyle p_1 = p$. Then:

$\displaystyle n^2 = p^{2a_1}\left(\prod_{i=2}^np_i^{2a_i}\right)$ ...etc

Also, when you reach the line:$\displaystyle n = p^{a_1}\left(\prod_{i=2}^np_i^{a_i}\right)$

I think you can say immediately that $\displaystyle p|n$.

Grandad