Relation between fully divide and deriviate of a known polynomial.

Question

I was working on an olympiad question of Japan. Where I have to prove that for any integer $n$, there is integer $x$ such that $v_3(x^3+27)=n$. My approach consist of Hensel lemma and inductive hypothesis. By Hensel's lemma, if there exist $a$ such that $3^k\mid f(a)$ and $3^k \not\mid f'(a)$. Then there is $b$ such that $3^{k+1}\mid f(b)$ and $b \equiv a \pmod {3^k}$. Since $b=a+n3^k$ so we could further deduce that $3^{k+1}\not\mid f'(b)$, so this induction continues. Then we prove the base case where $k=2$, in this case $a=1$. The case where $k=1$ is another case but its easy to see that a solution exist. Lemma 1: if $p^k \mid f(a)$ and $p^k \not\mid f'(a)$, then $p^k \mid\mid f(a)$ (Integer polynomial). But I was stuck on this lemma, I can't prove it true or find a counter example. If its not true is it true for $f(x)=x^3+17$?