Prove or disprove: there exists an integer $n$ such that $n^2 \equiv 2 \mod 3$.

Question

I'm having trouble with a homework problem due later today. I need to either prove or disprove the statement $$n^2 \equiv 2 \mod 3$$

I know that this can be written as $ n^2 = 3k + 2$, where $k \in \mathbb Z$.

I haven't been able to think of any examples, but neither do I know how to disprove this. Can anyone give me a hint on where to go from here?

Answer 1

For such a small modulus, we can simply check all the cases

\begin{align}0^2&\equiv 0\mod 3\\ 1^2 &\equiv 1\mod 3\\ 2^2 &\equiv 4\mod 3\equiv 1\mod 3\end{align}

Now we can see that $n^2 \not\equiv 2\mod 3$ in any case so there is no solution