Find all prime ideals in $\mathbb{Z}/n\mathbb{Z}$ where $n>1$

Another theorem says the prime ideals of $R/I$ correspond bijectively to the prime ideals of $R$ containing $I$.

In the case of $\mathbf Z/n\mathbf Z$, this means its prime ideals are generated by the congruence classes of the prime divisors of $n$.

If $J=(m)$ does not contain $n$, i.e. if $m$ does not divide $m$, the image of $J$ in $\mathbf Z/n\mathbf Z$ is the ideal $$J\cdot \mathbf Z/n\mathbf Z=(m)\cdot \mathbf Z/n\mathbf Z=(m,n)/(n)=(\gcd(m,n))/(n).$$