KöMaL 2017 A700

A positive integer $n$ satisfies the following: it is possible to select some integers such that if we randomly choose two different integers from this list, say, $i$ and $j$, then $i+j$ $\mathrm{mod\ } n$ is equal to one of the numbers $0,1,\dots,n-1$ with equal probability. Find all numbers $n$ with this property.