Let $f(x) = (x^2-13)(x^2-17)(x^2-221)$. For $n=2,3,5,7$ show there exists $x\in\mathbb Z$ such that $f(x) \equiv 0\pmod n$.
I am aware that this means that $f(x)$ is divisible by $n$ but am unsure how to go about this question. Do I expand the brackets? If so, then what?
This is true for all $n$ prime because if $13$ and $17$ are not quadratic residues mod $n$ then $221= 13 \cdot 17$ is a quadratic residue mod $n$. This follows from Euler's criterion.