  We continue the research of the relation
$\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ on the set $\beta {\mathbb{N}}$ of
ultrafilters on ${\mathbb{N}}$, defined as an extension of the divisibility
relation. It is a quasiorder, so we see it as an order on the set of
$=_\sim$-equivalence classes, where ${\cal F}=_\sim{\cal G}$ means that ${\cal
F}$ and ${\cal G}$ are mutually
$\hspace{1mm}\widetilde{\mid}\hspace{1mm}$-divisible. Here we introduce a new
tool: a relation of congruence modulo an ultrafilter. We first recall the
congruence of ultrafilters modulo an integer and show that $=_\sim$-equivalent
ultrafilters do not necessarily have the same residue modulo $m\in
{\mathbb{N}}$. Then we generalize this relation to congruence modulo an
ultrafilter in a natural way. After that, using iterated nonstandard
extensions, we introduce a stronger relation, which has nicer properties with
respect to addition and multiplication of ultrafilters. Finally, we also
introduce a strengthening of $\hspace{1mm}\widetilde{\mid}\hspace{1mm}$ and
show that it also behaves well in relation to the congruence relation.
