Past exam of the probability, statistics and options I course of the University of Oxford / 2009 / C / 1 / iv

Modulo $p$, the term $p^{q-1}$ is zero and Fermat's little theorem gives $q^{p-1}\equiv 1$, since $q\ne p$. Modulo $q$, the roles reverse. Hence the sum minus one is divisible by both distinct primes. Their coprimality gives