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Divisibility Tests: A History and User's Guide

Eric L. McDowell (Berry College)

A divisibility test is an algorithm that uses the digits of an integer \(N\) to determine whether \(N\) is divisible by a divisor \(d.\)  The history of divisibility tests dates back to at least 500 C.E. when a divisibility test for \(7\) was included in the Babylonian Talmud.  Since then, methods that provide divisibility tests for all positive integers have been discovered and rediscovered by a wide population of mathematicians and mathematical enthusiasts including Blaise Pascal [37], Joseph-Louis Lagrange [27], and Charles Dodgson (a.k.a. Lewis Carroll, author of Alice In Wonderland) [12].  An impressive summary of the literature regarding divisibility tests published prior to 1915 is provided in Leonard Dickson's History of the Theory of Numbers [10].  Edward Brooks devoted two chapters to the study of divisibility tests in his 1880 book The Philosophy of Arithmetic [5].  Much more recently (2006), Marc Renault published a delightful article [41] that provides divisibility tests for all integers between \(2\) and \(102,\) and which includes brief explanations for why the tests work.

The purpose of the present article is twofold.  The first is to present a modest survey of some of the more recent literature regarding divisibility tests while arguing that most all of these tests are actually rediscoveries of earlier known techniques.  The second is to describe how these techniques can be employed to produce a variety of divisibility tests for any integer.  We hope these investigations will serve to encourage readers to rediscover divisibility tests of their own.  More importantly, we hope readers will share these techniques of divisibility with their students to further their experience of the joy and enchantment of mathematical discovery.

Eric L. McDowell (Berry College), "Divisibility Tests: A History and User's Guide," Convergence (May 2018), DOI:10.4169/convergence20180513