BALTIC-WAY 1997 Problem 18

a) Prove the existence of two infinite sets $A$ and $B$, not necessarily disjoint, of non-negative integers such that each non-negative integer $n$ is uniquely representable in the form $n=a+b$ with $a \in A, b \in B$.

b) Prove that for each such pair $(A, B)$, either $A$ or $B$ contains only multiples of some integer $k>1$.