An Algebraic Approach To General Divisibility Rules

AbstractAn essential tool in number theory, divisibility rules have been documented for atleast the first 1000 prime numbers. However, as the divisors increase in magnitude,these per-number rules decline, leading us to depend on more generalized rules to coverevery number. Existing general algorithms for divisibility, such as osculation and thosebased on Pascal’s test for divisibility, are of high computational complexity and, assuch, quite difficult to scale. This paper introduces a generalized rule for all integersending with a particular digit in base 10, as well as a universally generalized rule that isapplicable to all integers in all bases. By deriving patterns based on a divisor’s terminaldigit and reversing what is typically used to prove such algorithms, I propose a singular,optimized divisibility rule of relatively simpler algebraic complexity and provide meansto increasing these rules’ efficiency.