Solve: $x^2\equiv 1 \pmod{20},x^2\equiv 6 \pmod{15},x^2\equiv 9 \pmod{18}.$ I want to solve: $x^2\equiv 1 \pmod{20}, x^2\equiv 6 \pmod {15}, x^2\equiv 9\pmod{18}.$ This is a system of congruence equations, but these are not linear and moduli are not coprime. So,we cannot apply chinese remainder theorem here. However, I think I can solve for $y\equiv 1 \pmod{20}, y\equiv 6 \pmod{15}, y\equiv 9 \pmod{18}$ by using extended chinese remainder theorem,then we have to search for the square numbers out of them.How to do that?

$$x^2 \equiv 1 \pmod{5} \iff x \equiv \pm 1 \pmod{5}$$
$$x^2 \equiv 1 \pmod{4} \iff x \equiv 1,3 \pmod{4}$$
$$x^2 \equiv 0 \pmod{9} \iff x \equiv 0 \pmod{3}$$
