elementary number theory - How many solutions does a quadratic congruence have?

Given $x^2 \equiv 182\ mod\ 727$, how many solutions $mod\ 727$ does it have? Note $727$ is prime and $182 = 2\cdot7\cdot13$.

So I know this is soluble computing $\left(\frac{182}{727}\right)$, but how do I determine the number of solutions? I know how to do this for linear congruences but not sure how to do it for quadratic.

Answer

$727\equiv 3\bmod 4$ so the computation of $$\pm 182^\frac{727+1}{4}=\pm182^{182}$$ in modulo $727$ will give you the desired solutions. Incidentally, the set of solutions are $\{363,364\}.$

As another user alludes in his comment; since $\Bbb Z_{727}$ is a field; the equation $x^2\equiv a\bmod 727$ will have at most two solutions.