A PEDAGOGICAL VISUALIZATION OF BASE 10 DIVISIBILITY CYCLES.

Divisibility is a fundamental concept in number theory, traditionally approached through arithmetic rules and modular computations. While these existing methods are rigorous, they often lack intuitive or visual representations, which can make it difficult to perceive repeating patterns in integer sequences. We introduce a two-layer cyclic diagram, called a pie chart, that represents known base 10 modular residue cycles of multiples of a fixed divisor 𝑑 in a geometric form designed for pedagogical exploration. The outer ring records unit-digit residues of successive multiples of d while the inner ring records changes in decimal carry behavior between consecutive multiples, providing a visual trace of decimal carry behavior. Together these rings form a finite cyclic diagram that visually organizes base 10 modular residue behavior relative to 𝑑. We demonstrate that the unit digit residues of multiples of any positive integer 𝑑 always form a finite repeating cycle, which can be compactly visualized in this diagram. The framework supports intuitive exploration of residue periodicity divisibility, computational visualization of modular patterns and pedagogical analysis of arithmetic sequences.