Let $p$ be a prime number and let $n$ be such that $p^k \nmid n$. Prove: If the equation $y^n\equiv a \pmod {p^k}$ has a solution with $\mathrm {gcd} (y,p)=1$, then for every $m>k$, there is a solution for the equation $y^n\equiv a \pmod {p^m}$.
This is a problem from an introductory course in number theory, and is in the Hensel's Lemma section, which contains the generalized version where $p^l$ can divide the derivative if $2l< k$ . I tried to modify the proof for Hensel's Lemma, but I couldn't deal with the lack of the above hypothesis.