Prime elements in $\mathbb{Z}/n\mathbb{Z}$ I'm tring to determine the prime elements in the ring $\mathbb{Z}/n\mathbb{Z}$.   

Hint $\rm\:\ (p,n) = 1\!\!\!\overset{\rm Bezout\!\!}\iff p\:$ is a unit in $\rm\: \Bbb Z/n.\:$ Else $\rm\:\color{#c00}{p\:|\:n}\:$ therefore
$$\rm\: p\:|\: ab\ in\ \Bbb Z/n\:\Rightarrow\: \color{#c00}pq = ab + k\color{#c00}n\ in\ \Bbb Z\:\Rightarrow\:p\:|\:ab\ in\ \Bbb Z\:\Rightarrow\:p\:|\:a\,\ or\,\ p\:|\:b\ in\ \Bbb Z,\ so\ also\ in\ \Bbb Z/n$$
This is aspecial case of a general correspondence principle between prime ideals in a ring R and its quotient rings.
