A new necessary condition for moduli of non-natural irreducible disjoint covering system.
I. Polách · 1994
. A disjoint covering system S = (a1 (mod n1 ); : : : ; a k (mod n k )) is said to be irreducible if the union of any of its r residue classes, 1 ! r ! k, is not a residue class. An irreducible disjoint covering system is non-natural if not all its moduli are equal. The least common multiple of its moduli n 1 ; : : : ; n k will be called the common modulus of S. The main and most interesting result of this paper is Theorem 2.2 giving this neccesary condition: if p ff is a divisor of the common modulus of S (p a prime), then there exist at least 3 residue classes in S with the pairwise different moduli divisible by p ff . In the last section an example class of irreducible systems with the set of moduli containing exactly 4 elements is given. 1. Introduction and Basic Properties Denote Z the set of all integers. By symbols gcd and lcm we mean the greatest common divisor and the least common multiple respectively. For any integers n ? 0 and a the symbol a (mod n) will denote the re...