  This study investigates a multiplicative integer system using a method that
was developed for studying pattern generation problems. The entropy and the
Minkowski dimensions of general multiplicative systems can thus be computed. A
multi-dimensional decoupled system is investigated in three main steps. (I)
Identify the admissible lattices of the system; (II) compute the density of
copies of admissible lattices of the same length, and (III) compute the number
of admissible patterns on the admissible lattices.
  A coupled system can be decoupled by removing the multiplicative relation set
and then performing procedures similar to those applied to a decoupled system .
The admissible lattices are chosen to be the\ maximum graphs of different
degrees which are mutually independent. The entropy can be obtained after the
remaining error term is shown to approach zero as the degree of the admissible
lattice tends to infinity.
