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$\beta {\mathbb {Z}}$
We introduce self-divisible ultrafilters, which we prove to be precisely those 
$w$ such that the weak congruence relation 
$\equiv _w$ introduced by Šobot 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 
$\mathrel {\equiv ^{\mathrm {s}}_{w}}$, if and only if the quotient 
$(\beta {\mathbb Z},\oplus )/\mathord {\mathrel {\equiv ^{\mathrm {s}}_{w}}}$ 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.
