  This article analyzes the periodic orbits of Syracuse dynamical systems in a
novel algebraic setting: the commutative ring of graded $n$-adic integers.
Within this context, this article introduces a dual-radix modular division
algorithm for computing the graded canonical expansions and graded quotients
for a certain class of rational expressions that arise from periodic orbits
within these dynamical systems. This division algorithm yields two novel
methods for testing the integrality of the B\"{o}hm-Sontacchi numbers.
