= Solution
Use the <hook criterion in a beta set>: for a finite <beta set of a partition> $X$, each <hook length> is a difference $b-c$ with $b\in X$, $0\leq c<b$ and $c\notin X$, and every such difference is a hook length.
A hook length $pr$ therefore gives a bead $b$ and a gap $c=b-pr$. Along the progression
$$
b,\ b-p,\ b-2p,\ldots,\ b-rp=c,
$$
the first position is a bead and the last is a gap. Some adjacent positions are consequently a bead followed by a gap. Their difference is $p$, so $p$ is a <hook length>. Similarly the progression from $b$ to $c$ in steps of $r$ gives a hook length $r$. Every position used is nonnegative because it lies between $c$ and $b$.
Thus the <divisor closure of hook lengths> yields
$$
\boxed{pr\in\mathcal H(\lambda)\ \Longrightarrow\ p,r\in\mathcal H(\lambda).}
$$
Neither $p$ nor $r$ needs to be a <prime number>.
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