Problem - 4156
Let $n^2$ be a square number. Show that $n^2\equiv 0, \pm 1\pmod{5}$.
If $n\equiv 0\pmod{5}$, then $n^2\equiv 0\pmod{5}$.
If $n\equiv \pm 1\pmod{5}$, then $n^2\equiv 1\pmod{5}$.
If $n\equiv \pm 2\pmod{5}$, then $n^2\equiv 4 \equiv -1\pmod{5}$.
Therefore, we conclude the claim holds.