Question

If $p$is prime and $p|a^n$, is it true that $p^n|d^n$? Justify your answer.

[Hint: Corollary 1.6.]

Short answer

Yes, it is true that$p^n|d^n$

Step-by-step solution

Use the given part

It is given that $p$is a prime and $p|a^n$.

This implies that the powers of $p$also divides $a^n$, that is $p|a^n$.

Proof

Write $a^n$as $a^n=a{a}^{n-1}$.

So, if $p$divide $a^n$, then $p|a$or $p|a^{n-1}$. Hence, if $p|a$,then $p^n|a^n$.

If $p|a^{n-1}$, then $a^{n-1}=a{a}^{n-2}$that gives $p|a^{n-2}$,counting as $n-1$. Then again $p|a$implies $p^n|a^n$.