<s>[INST] Perfect squares and modularly congruency in mod 5 There is not any perfect square $k$ such that $k  \equiv 3\ (\textrm{mod}\ 5)$? Why? How can I prove it?
 [/INST] You can check the 5 cases:
If $n \equiv 0 \pmod{5} \Rightarrow n^2 \equiv 0 \pmod{5}$
If $n \equiv 1 \pmod{5} \Rightarrow n^2 \equiv 1 \pmod{5}$
If $n \equiv 2 \pmod{5} \Rightarrow n^2 \equiv 4 \pmod{5}$
If $n \equiv 3 \pmod{5} \Rightarrow n^2 \equiv 4 \pmod{5}$
If $n \equiv 4 \pmod{5} \Rightarrow n^2 \equiv 1 \pmod{5}$
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