= Solution
\b[No.] The ten <residue classes> modulo $10$ form a finite partition, so an <ultrafilter> $p$ contains exactly one class $C_r$. The addition formula shows that $p+p$ contains $C_{2r}$: for $n\in C_r$, the translate $C_{2r}-n$ is $C_r$ and belongs to $p$. If $p$ is an <idempotent ultrafilter>, uniqueness of its selected class gives $2r\equiv r\pmod{10}$, hence $r=0$.
Thus \b[every <idempotent ultrafilter> contains the multiples of $10$], and cannot also contain their complement. This is the <zero-residue constraint for idempotent ultrafilters>, valid for every finite modulus.
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