1. Number Theory

Prove or Disprove the following statements

If a,b in Z bot greater than 0, and a^2|b^3, than a|b
If a in Z, a>0, p is prime, and p^4|a^2, then p^2|a.

2. 1. If $b=n^{2}$ and $a=n^3$ for any integer $n\neq{0}$ and $n\neq{\pm{1}}$then $a^2=b^3$ so $a^2|b^3$ but obviously $a|b$ doesn't hold

2. This is true.By Euclid's Lemma p|a. Suppose $
p^s
$
is the max. power of p dividing $a$, since $p^4|a^2$ it follows that $2s\geq{4}$ so $s\geq{2}$, therfore $p^2$ divides $a$(since s is greater or equal than 2)

3. Counterexample for first problem: a = 8, b = 4.