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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 $b\in X$, $0\le c<b$ 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 progressionthe 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 yieldsNeither $p$ nor $r$ needs to be a prime number.

The supplied row-hook formula isConsequently $h\in H_i(\lambda )$ exactly when $h_i-h\ge 0$ and $h_i-h$ is not one of $h_{i+1},\dots ,h_m$. Since $h_i-h\le h_i<h_j$ for $j<i$, this is equivalent to $h_i-h\notin X$. We have proved the hook criterion in a beta set

If $ef$ is a hook length, the beta-set interpretation gives a bead at some position $b$ and a gap at $b-ef$. In the finite progressionthe first position is occupied and the last is empty. Some consecutive pair is therefore a bead followed by a gap. Their distance is $e$, so the criterion gives a hook of length $e$. This proves the divisor closure of hook lengths.