Perfect square is $0$ or $1$ modulo $4$ .

Question

Prove that for every integer $n$ either

• $n^2 \equiv 0\pmod{4}$ or

• $n^2\equiv 1\pmod{4}$

Answer 1

Hint:

If $n = 2k$, $$n^2 = 4k^2$$ If $n = 2k+1$, $$n^2 = (2k+1)^2 = 4k^2+4k+1 = 4(k^2+k)+1$$

Answer 2

$n$ can be even or odd.

When $n=2k$ we get

$(2k)^2=4k^2 \equiv 0 \pmod 4$.

When $n=2k+1$ we get

$n^2 = (2k+1)^2 = 4k^2 + 4k + 1 \equiv 1 \pmod 4$.