On the reducibility of exact covering systems

Abstract

There exist irreducible exact covering systems (ECS). These are ECS which are not a proper split of a coarser ECS. However, an ECS admiting a maximal modulus which is divisible by at most two distinct primes, primely splits a coarser ECS. As a consequence, if all moduli of an ECS $A$, are divisible by at most two distinct primes, then $A$ is natural. That is, $A$ can be formed by iteratively splitting the trivial ECS.