Past exam of the mathematics course of the University of Cambridge / 2024 / ia / Paper 4 / 1E / Solution

The Wilson theorem states that for a prime $p$,Indeed, every nonzero residue modulo $p$ has a unique multiplicative inverse. The only residues equal to their own inverses solve $x^2\equiv 1(\mathrm{mod}p)$, hence are $1$ and $-1$. Pairing every other residue with its distinct inverse leaves

The Fermat little theorem states that for prime $p$,equivalently, if $p\nmid a$, then $a^{p-1}\equiv 1(\mathrm{mod}p)$.

Wilson's theorem at $p=31$ givesso $28!\equiv 15(\mathrm{mod}31)$. Also $729\equiv 16(\mathrm{mod}31)$, and therefore