Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?
Problem
Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade
Tags: number theory, prime divisors, prime numbers, Divisibility