DE-BUNDESWETTBEWERB 2024 Problem 4

For positive integers $p$ , $q$ and $r$ we are given $p \cdot q \cdot r$ unit cubes. We drill a hole along the space diagonal of each of these cubes and then tie them to a very thin thread of length $p \cdot q \cdot r \cdot \sqrt{3}$ like a string of pearls.

We now want to construct a cuboid of side lengths $p$ , $q$ and $r$ out of the cubes, without tearing the thread. a) For which numbers $p$ , $q$ and $r$ is this possible? b) For which numbers $p$ , $q$ and $r$ is this possible in a way such that both ends of the thread coincide?