On exact coverings of the integers

By an exact covering of modulusm, we mean a finite set of liner congruencesx≡a _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 $$mathop Sigma limits_{d/m} mu (d)left( {Tleft( {frac{m}{d}} right)} right)^d = 1$$ .