PLease help me with this:
If p is prime and p|a^n, is it true that p^n|a^n? Justify your answer.
Thank you.
Yes indeed. If $\displaystyle p|a^n \Rightarrow p|a$. You can show this inductively:
It follows from the prime-property: $\displaystyle p|xy \Rightarrow p|x$ or $\displaystyle p|y$. Take $\displaystyle x= a, y= a^{n-1}$. If $\displaystyle p|a$ you're done if $\displaystyle p|a^{n-1}$ you go on with $\displaystyle x= a, y= a^{n-2}$. Eventually you'll obtain $\displaystyle p|a$
Conclusion: $\displaystyle p^n|a^n$