Solution
= Solution
The converse is \b[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>, are
$$
\begin{matrix}
4&3&2\\
3&2&1
\end{matrix}.
$$
Both $2$ and $3$ occur, but $6$ does not, even though $pr=6=n$. Hence
$$
\boxed{\lambda=(3,3),\quad p=2,\quad r=3}
$$
is a counterexample with distinct $p,r$.