Past exam of the mathematics course of the University of Cambridge / 2026 / ia / Paper 4 / 6D / iii / Solution

For prime $n$, pair each nonzero residue with its inverse; only $\pm 1$ are self-inverse, yielding Wilson theorem. Conversely, if $(n-1)!\equiv -1(\mathrm{mod}n)$, every $1\le a<n$ is a unit (otherwise a common divisor would divide the left side and $n$ but not $-1$), so $n$ is prime.