Let $m =\prod_{i=1}^{r} p_i^{α_i},$ with $α_i \ge 1$ and $p_i \ge 3$ for each $i$, be the canonical representation of $m$ and let $a$ be relatively prime to $m$. Show that $x^2 \equiv a \pmod m$ is solvable if and only if $\frac{a}{p_i} = 1$ for each $i$.
-Having a tough time with this question and came back to my professor who's hint was to solve for the power of the prime via Chinese Remainder Theorem and to use Hensel's "lifting" Lemma. Unfortunately this was a bit unclear to me and I am still having a rough time with this one, any help is greatly appreciated.
-Also I am under the assumption that this solvable because the next question on the review asks to prove that there exist precisely $2^r$ incongruent solutions modulo $m$, given that its solvable of course.