Past exam of the mathematics course of the University of Cambridge / 2020 / ii / Paper 4 / 10C / iii / Solution

Applying the transform twice givesBy the finite geometric series, the inner sum over $y$ is $N$ when $x+z\equiv 0(\mathrm{mod}N)$ and zero otherwise. Thereforewhich is the stated $|N-x⟩$ with the label interpreted modulo $N$.