4 Prove that if n is an integer then n2 0 or 1 mod 4 Solutio

4. Prove that if n is an integer, then n^2 0 or 1 (mod 4).

Solution

Let us assume n = (even integer) = 2*k (where k is an integer)

(2k)^2 mod 4 = 4k^2 mod 4 = 0

Now let us assume n = (odd integer) = 2*p + 1, (where p is an integer)

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

=> (4p^2 + 4p) mod 4 + 1 mod 4

=> 0 mod 4 + 1 mod 4

=> 1 mod 4

Hence n^2 will be either 0 mod 4 in the case of (n=even) or 1 mod 4 in the case of (n=even)