On exact coverings of the integers

AbstractBy an exact covering of modulusm, we mean a finite set of liner congruencesx≡ai (modmi), (i=1,2,...r) with the properties: (I)mi∣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=pαqβ and letT(m) be the number of exact coverings of modulusm. Then,T(m) is given recursively by $$\mathop \Sigma \limits_{d/m} \mu (d)\left( {T\left( {\frac{m}{d}} \right)} \right)^d = 1$$ .