Israel Journal of Mathematics

, Volume 12, Issue 3, pp 299–305 | Cite as

On exact coverings of the integers

  • John Friedlander


By an exact covering of modulusm, we mean a finite set of liner congruencesxa i (modm i ), (i=1,2,...r) with the properties: (I)m i m, (i=1,2,...,r); (II) Each integer satisfies precisely one of the congruences. Let α≥0, β≥0, be integers and letp andq be primes. Let μ (m) senote the Möbius function. Letm=p α q β and letT(m) be the number of exact coverings of modulusm. Then,T(m) is given recursively by
$$\mathop \Sigma \limits_{d/m} \mu (d)\left( {T\left( {\frac{m}{d}} \right)} \right)^d = 1$$


Nontrivial Solution Trivial Solution Recursion Formula Residue Class Chinese Remainder Theorem 
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  1. 1.
    P. Erdös,On a problem concerning congruences systems, Mat. Lapok.3 (1952), 122–128.MathSciNetGoogle Scholar
  2. 2.
    J. H. Jordan,Convering classes of residues, Canad. J. Math.3 (19), (1967), 514–519.Google Scholar

Copyright information

© Hebrew University 1972

Authors and Affiliations

  • John Friedlander
    • 1
  1. 1.The Pennsylvania State UniversityUniversity Park

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