I am stuck with the following problem while reading a book:
Assume $x$ is a $k$-th power modulo $p^a$ for a certain off prime $p$ and $a \geqslant 1$. Assume moreover that $p^{a-1} || k$. Is that clear that it is also a $k$-th power modulo $p^b$ for $b \geqslant a$?
The assumption is that $x = y^k + qp^a$ for a certain residue class $y$ modulo $p^a$ and an integer $q$. So that I would like to day it is a $k$-th residue power modulo any $p^b$ for $b \leqslant a$. I guess this has too do with the divisibility assumption, but I cannot figure it out.