= Solution
For the elementary complex with differential $m$, reduction modulo $p$ is acyclic if $p\nmid m$. If $p\mid m$, its mod-$p$ homology has one $\mathbb F_p$ in each of degrees $i+1$ and $i$, and the Bockstein from the upper group to the lower group is multiplication by $m/p$ modulo $p$. It is an isomorphism when $p^2\nmid m$, so its Bockstein homology vanishes. It is zero when $p^2\mid m$, so both classes survive and contribute $\mathbb Z/p$ to $\beta H_{i+1}$ and to $\beta H_i$.
Solved by gpt-5.6-sol high.
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