  We introduce self-divisible ultrafilters, which we prove to be precisely
those $w$ such that the weak congruence relation $\equiv_w$ introduced by
\v{S}obot is an equivalence relation on $\beta\mathbb{Z}$. We provide several
examples and additional characterisations; notably we show that $w$ is
self-divisible if and only if $\equiv_w$ coincides with the strong congruence
relation $\equiv^{\mathrm{s}}_{w}$, if and only if the quotient
$(\beta\mathbb{Z},\oplus)/\mathord{\equiv^{\mathrm{s}}_w}$ is a profinite
group. We also construct an ultrafilter $w$ such that $\equiv_w$ fails to be
symmetric, and describe the interaction between the aforementioned quotient and
the profinite completion $\hat{\mathbb{Z}}$ of the integers.
