Past exam of the options I course of the University of Oxford / 2009 / C / C1 / iv / Solution

Because the primes are distinct, Fermat's little theorem gives $q^{p-1}\equiv 1(\mathrm{mod}p)$ and $p^{q-1}\equiv 1(\mathrm{mod}q)$. The other term in each modulus vanishes. ThereforeThe distinct primes are coprime, so their product divides this integer. Hence