  We introduce a class of graphs called compound graphs, generalizing
rectangles, which are constructed out of copies of a planar bipartite base
graph. The main result is that the number of perfect matchings of every
compound graph is divisible by the number of matchings of its base graph. Our
approach is to use Kasteleyn's theorem to prove a key lemma, from which the
divisibility theorem follows combinatorially. This theorem is then applied to
provide a proof of Problem 21 of Propp's Enumeration of Matchings, a
divisibility property of rectangles. Finally, we present a new proof, in the
same spirit, of Ciucu's factorization theorem.
