I have trouble in solving a basic exercise of the book Number Theory by Shafarevich and Borevich. It is exercise 4, chapter 1, page 4 in my edition.
It goes as follows: Using the properties of the Legendre-Symbol, show that the congruence $$(x^2-13)(x^2-17)(x^2-221) \equiv 0 \mod m$$ is solvable for all $m$.
I know how to deal with the problem if $m$ is a prime. However, I don't know what to do if $m$ is a power of a prime. (I also know that solving the problem for all prime powers suffices.)
The idea is that if neither of $13$ or $17$ are a quadratic residue $\mod p^n$ then $13 \cdot 17 = 221$ must be a quadratic residue (this follows from the group of units of $\mathbb{Z}/p^n \mathbb{Z}$ being cyclic for odd $p$ and you need to consider the case $p = 2$ separately)