**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. 63, 1 (1994)

pp. 133-140

A NEW NECESSARY CONDITION FOR MODULI OF NON-NATURAL IRREDUCIBLE DISJOINT COVERING SYSTEM

I. POLACH

**Abstract**.
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, $1<r<k$, 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 $n_1, \dots, n_k$ 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.

**AMS subject classification**.

**Keywords**.
Disjoint covering system, irreducible disjoint covering system

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