Find the number of solutions to $x^2 ≡ 1\bmod(990)$.
Hi everyone, is there a clever systematic way to calculate all the answers, or the number of solutions, or must you use brute force and calculate all squares of $Z_{990}$?I cheated and got all solutions with a little help from a friend (Excel) and got $±1$, $±89$, $±109$ and $±199$. However I know that the primfactorization of $990$ is $2\times3^2\times5\times11$, does it help me to know that $Z_{990} \cong(Z_2 \times Z_9 \times Z_5 \times Z_{11})$ and if so, how? Any help is appreciated.
\begin{align} x & \equiv 1 & & \pmod{2} \\ x & \equiv \pm 1 & & \pmod{5} \\ x & \equiv \pm 1 & & \pmod{9} \\ x & \equiv \pm 1 & & \pmod{11} \\ \end{align} Solve the $8$ possibilities above using the the Chinese remainder theorem.