Question about primes in $\mathbb{Z}_{n} $

Question

For example, I kind of expect that in $\mathbb{Z}_{6}$ $\overline{2}$ is prime but I don't know how to prove it (considering the fact that I can test all the different products, but I want a more theorical proof), because I want to show that $\left(\overline{2}\right)$ is a prime ideal. Thanks in advance!