Verified answer

Let n be odd. Then n = 2k + 1. Then n^2 = 4k^2 + 4k + 1. Then 4(k^2 + k) = n^2 - 1. Thus, 4 | (n^2 - 1). Thus, n^2 = 1 (mod 4).

The first case n=1 works

All positive odd integers are of the form 2x+1

(2x+1)^2(mod 4)=4x^2+4x+1(mod 4)

4x^2+4x is divisbile by 4 so 4x^2+4x+1 (mod 4)≡1

n is odd so n= 2k+ 1

so n^2 = 4k^2+ 4k + 1 = 4(k^2 + k) + 1

so n mod 4 = 1