Prove if n is an integer that is not a multiple of 3 then n2

Prove: if n is an integer that is not a multiple of 3, then n^2 equiv 1mod 3.

n is not a multiple of 3 (given)

this means n+1 and n-1 both are multiples of 3

thus n+1=0 mod 3

and n-1=0 mod 3

Also (n+1)(n-1) would be a multiple of 3 (simple maths logic)

Hence (n-1)(n+1)= 0 mod 3

implies n^2-1=0 mod 3 (using the identity a^2-b^2=(a+b)(a-b))

thus n^2-1 is a multiple of 3 which means n^2-1+1 will leave a remainder 1 when divided by 3

Thus n^2=1 mod 3