Zero-residue constraint for idempotent ultrafilters ($p\text{ selects }0(\mathrm{mod}q)$)

An ultrafilter on the positive integers selects exactly one residue class modulo a fixed positive integer $q$. Under addition on the Stone-Čech compactification of the natural numbers, selected residues add. An idempotent ultrafilter must therefore select a residue $r$ with $2r=r$ in the finite cyclic group, hence $r=0$. In particular, the multiples of every fixed positive integer belong to every additive idempotent ultrafilter.