Prove that for every integer n either n2 3 horizontal lines

Prove that for every integer n, either n^2 (3 horizontal lines, similar to an = sign) 0(mod4) or n^2 (3 horizontal lines) 1(mod 4)

Solution

Let the number n being an even number, then n can be written as

n = 2p

n^2 = 4p^2

4p^2 mod(4) = 0

Hence n^2 mod(4) = 0

For n being odd

n = 2q+1

n^2 = (2q+1)^2 = 4q^2 + 1 + 4q

n^2 mod(4) = 1

Hence for n being an even number it will be 0 mod 4 and for n being odd, it will be 1 mod 4