Exact coverings of the integers by arithmetic progressions

For a given M we may ask questions analogous to I and II above, but applied to ECS(M)'s rather than ordinary covering systems. It i s the purpose of this paper to provide partial answers to these questions. In Section 4 we provide upper and lower bounds on the prime divisors of the moduli of an ECS(M) , and an upper bound on the least modulus occurring in an ECS(M). Section 3 of the paper will be devoted to the proof of our central result, which is as follows.