$pq\mid pn^q+qn^p+kn$ for some $k,\,$ if $p,q$ are primes

Question

I get some troubles with this problem, could you help me?
Given prime numbers $p$ and $q$ prove $\exists k\in\mathbb{Z}$ such that $pn^q+qn^p+kn$ is a multiple of $pq,~\forall n\in\mathbb{Z}$.

Answer 1

Hint $\ $ Write it as $\ p(n^{\large q}\!-n)+q(n^{\large p}\!-n) +\!\! \overbrace{(p\!+\!q+k)}^{\large =\ 0\ \ {\rm if}\ \ k\ =\ \ldots}\!\!\!n\ $ and apply little Fermat