Question 163
Find all positive integers $n$ with the following property: there exist nonempty finite sets of integers $A, B$, such that for every integer $m$, exactly one of the following three statements is true,
(i) there exists $a \in A$, such that $m \equiv a \pmod{n}$;
(ii) there exists $b \in B$, such that $m \equiv b \pmod{n}$;
(iii) there exist $a \in A$ and $b \in B$, such that $m \equiv a+b \pmod{n}$.
(i) there exists $a \in A$, such that $m \equiv a \pmod{n}$;
(ii) there exists $b \in B$, such that $m \equiv b \pmod{n}$;
(iii) there exist $a \in A$ and $b \in B$, such that $m \equiv a+b \pmod{n}$.