Tiling the integers with translates of one finite set
  A set is said to tile the integers if and only if the integers can be written as a disjoint union of translates of that set. We consider the problem of finding necessary and sufficient conditions for a finite set to tile the integers. For sets of prime power size, it was solved by D. Newman [J. Number Theory 9 (1977), 107--111]. We solve it for sets of size having at most two prime factors. The conditions are always sufficient, but it is unknown whether they are necessary for all finite sets. 
Ethan M. Coven, Aaron D. Meyerowitz