CAPS 2018 Problem 6

We say that a positive integer $n$ is fantastic if there exist positive rational numbers $a$ and $b$ such that $$n = a + \frac 1a + b + \frac 1b.$$ (a) Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic.(b) Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic.

Proposed by Walther Janous, Austria