Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$$$n + 2 | n^p + q^p.$$Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.
Source: Switzerland - 2017 Swiss MO Final Round p4
Tags: number theory, divides, divisible
Let $n$ be a natural number and $p, q$ be prime numbers such that the following statements hold: $$pq | n^p + 2$$$$n + 2 | n^p + q^p.$$Show that there is a natural number $m$ such that $q|4^mn + 2$ holds.