n*n mod 3

Let N = ( 3p + k) , here k is either 0 , 1 or 2. As the number N is not divisible by 3 k =1 or 2. So N N = 9p p + 6pk + k k ; As it can be easily seen the first two terms are divisible by 3, the last term is k k is either 1 or 4 both when divided by 3 will leave a reminder of 1.

We can always express an integer indivisible by 3 as $N=3n\pm 1$ , where $n$ is an integer. Then, we have:

$\begin{aligned} N^2 & \equiv (3n\pm 1)^2 \equiv (0\pm 1)^2 \equiv \boxed 1\text{ (mod 3)}\end{aligned}$

Note that $N$ can be either positive or negative.

Due to the nature of numerical answer problems, you need to look at the phrase "what is _ mod 3." Since there can only be three answers, and the problem accepts a maximum of 3 answers, simply go through each of the possibilities: 0, 1, and 2. This problem would be much better as a multiple choice.