An Algebraic Approach To General Divisibility Rules

Abstract
An essential tool in number theory, divisibility rules have been documented for at
least 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 cover
every number. Existing general algorithms for divisibility, such as osculation and those
based on Pascal’s test for divisibility, are of high computational complexity and, as
such, quite difficult to scale. This paper introduces a generalized rule for all integers
ending with a particular digit in base 10, as well as a universally generalized rule that is
applicable to all integers in all bases. By deriving patterns based on a divisor’s terminal
digit and reversing what is typically used to prove such algorithms, I propose a singular,
optimized divisibility rule of relatively simpler algebraic complexity and provide means
to increasing these rules’ efficiency.