Geodesic
A NEW NECESSARY CONDITION FOR MODULI OF NON-NATURAL IRREDUCIBLE DISJOINT COVERING SYSTEM
Acta mathematica Universitatis Comenianae, Tome 63 (1994) no. 1 
I. Polach. A NEW NECESSARY CONDITION FOR MODULI OF NON-NATURAL IRREDUCIBLE DISJOINT COVERING SYSTEM. Acta mathematica Universitatis Comenianae, Tome 63 (1994) no. 1. http://geodesic.mathdoc.fr/item/AMUC_1994_63_1_a7/
@article{AMUC_1994_63_1_a7,
     author = {I. Polach},
     title = {A {NEW} {NECESSARY} {CONDITION} {FOR} {MODULI} {OF} {NON-NATURAL} {IRREDUCIBLE} {DISJOINT} {COVERING} {SYSTEM}},
     journal = {Acta mathematica Universitatis Comenianae},
     year = {1994},
     volume = {63},
     number = {1},
     url = {http://geodesic.mathdoc.fr/item/AMUC_1994_63_1_a7/}
}
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A disjoint covering system $\s =\left(a_1\pmodn_1, \dots, a_k\pmodn_k \right)$ is said to be irreducible if the union of any of its $r$ residue classes, $1, 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 $n_1, \dots, n_k$ 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 \pa 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 \pa. In the last section an example class of irreducible systems with the set of moduli containing exactly 4 elements is given.