Let A, B ⊆ N be two finite sets of natural numbers. We say that B is an additive divisor of A if there exists some C ⊆ N with A = B + C, where ‘+’ denotes the sumset operation, also called Minkowski sum or pairwise sum. We prove that among subsets of {0,1,...,k} containing 0 as an element, the full set {0,1,...,k} has the most additive divisors. To remove the restriction of having 0 as an element, we first prove a correspondence between the sumset operation and binary lunar multiplication. Lunar arithmetic (originally named "dismal arithmetic") is a kind of min/max arithmetic introduced by Applegate, LeBrun, and Sloane. The number of binary lunar divisors is related to restricted compositions of integers—restricted in that the first part must be greater or equal to all other parts. We establish some bounds on the number of such compositions and conclude that {1,...,k} has the most additive divisors among all subsets of {0,1,...,k}. These two results prove two conjectures of LeBrun et al. regarding the maximal number of lunar binary divisors, a special case of a more general conjecture about lunar divisors in arbitrary bases. We prove this third conjecture by introducing a sumset-like operation for multisets.