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

Pattern generation problems arising in multiplicative integer systems