Constructions of Coverings of the Integers: Exploring an Erdős Problem

In this paper, we study necessary conditions for small sets of congruences with distinct moduli to cover the integers, and we construct larger covering systems that address a problem of Erdős: “What is the largest minimum modulus needed for a set of congruences with distinct moduli to cover the integers? ” We show that the fewest number of distinct moduli necessary to cover the integers is 5, and that there is only one set of distinct moduli with which to construct a covering with only 5 congruences. We determine which natural numbers less than 50 can be the least common multiple of the moduli of a covering. We establish that the minimum modulus for a covering system of distinct moduli is at least 11.