Find all soultions of $\displaystyle x^2 \equiv$ -23 ( mod 240).
240 is a multiple of 5, and $\displaystyle -23\equiv2\!\!\!\pmod5$. But the equation $\displaystyle x^2\equiv2\!\!\!\pmod5$ has no solutions. So it doesn't look as though $\displaystyle x^2\equiv-23\!\!\!\pmod{240}$ has any solutions either.