Problem: Prove that $F(x)=(x^2-17)(x^2-19)(x^2-323)\equiv 0 \pmod{p^j}$ is solvable for all $p$ prime and $j\in\mathbb{N}$.
My attempt: With the help of Euler's criterion, I was able to prove that $F(x)\equiv0 \pmod{p}$ for any prime $p$. Then the hint given for this problem was to use Hensel's lemma to prove the existence of solutions to $F(x)\equiv0\pmod{p^j}$ for $j\geq2$. But I had trouble applying Hensel's lemma because, for example, $$2\mid F'(x)=2x(x^2-19)(x^2-323)+2x(x^2-17)(x^2-323)+2x(x^2-17)(x^2-19),\forall x\in\mathbb{Z}.$$ so I can't use Hensel's lemma to guarantee a solution for $F(x)\equiv0\pmod{2^j}$, $j\geq2$.
Could anyone suggest a solution?
You don't apply Hensel's lemma to the whole $F$.
Instead, try to apply it to each factor.
More precisely, for any odd prime number $p$, one of the three factors $x^2 - 17$, $x^2 - 19$, $x^2 - 323$ admits a root modulo $p$ (this is what you've got).
You then ask Hensel to give you a root modulo $p^j$ for all $j > 0$.
The case $p = 2$ is of course special, but obvious.