Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 103 / 2 / ii / Solution

The converse is false. Take $p=2$, $r=3$, $n=6$ and the partition of an integer $\lambda =(3,3)$. Its hook lengths, arranged in its Young diagram, areBoth $2$ and $3$ occur, but $6$ does not, even though $pr=6=n$. Henceis a counterexample with distinct $p,r$.