Past exam of the mathematics course of the University of Cambridge / 2019 / ia / Paper 4 / 5E / a / Solution

Fermat's little theorem states that for a prime $p$ and integer $a$,if $p\nmid a$, equivalently $a^{p-1}\equiv 1(\mathrm{mod}p)$. For the latter form, multiplication by $a$ permutes the nonzero residue classes modulo $p$. Multiplying the congruences $a,2a,\dots ,(p-1)a$ and cancelling $(p-1)!$ gives $a^{p-1}\equiv 1$. The first form also covers $p\mid a$.

Since $803=50\cdot 16+3$, the theorem gives