Determine whether $x^2 - 14x + 30 \equiv 0$ mod 1615 is solvable. If so, find its solutions... Determine whether $x^2 - 14x + 30 \equiv 0\pmod{1615}$ is solvable. If so, find its solutions.
I assume the best way to solve this is via Chinese Remainder Theorem, but first i would have to break down the mod and if there is a solution then utilize C.R.T. No quite sure how to attack this with the right numbers, any hints/help are very appreciated.

$$x^2-14x+30\equiv 0\pmod{5\cdot 17\cdot 19}$$
$$\iff (x-7)^2\equiv 19\pmod{5\cdot 17\cdot 19}$$
$$\iff \begin{cases}(x-7)^2\equiv 19\equiv 2^2\pmod{5}\\(x-7)^2\equiv 19\equiv 6^2\pmod{17}\\(x-7)^2\equiv 19\equiv 0^2\pmod{19}\end{cases}$$
$$\iff \begin{cases}x-7\equiv \pm 2\pmod{5}\\x-7\equiv \pm 6\pmod{17}\\x-7\equiv 0\pmod{19}\end{cases}$$
I used the fact that $5,17,19$ are prime and Euclid's Lemma: e.g., if $5\mid (x-7)^2-2^2=((x-7)+2)((x-7)-2)$, then $x-7\equiv \pm 2\pmod{5}$.
Now use Chinese Remainder Theorem to find all the $4$ solutions.
