Thus we conclude that in fact 95733553 is a multiple of 67.
This article has two aims. First, to identify six categories of tests that most divisibility tricks fall into, and second, to provide an easy divisibility test for each number from 2 – 102 (thus the “101 Ways...” in the title). We’ll see that in fact many numbers have more than one divisibility test.
Divisibility tests have always fascinated people. Many of us learn “the rule of three” in childhood: a number is divisible by 3 if and only if the sum of its digits is divisible by 3. The Babylonians knew that a number of the form 100a + b is divisible by 7 if and only if 2a + b is divisible by 7. Chapter 12 of L. E. Dickson’s classic 1919 text History of the Theory of Numbers is entitled “Criteria for Divisibility by a Given Number” and contains a collection of divisibility tests gathered throughout history and covering many cultures. In a 1962 Scienti c American article Martin Gardner discusses divisibility rules for 2 – 12, and he explains that the rules were widely known during the Renaissance and used to reduce fractions with large numbers down to lowest terms. Today, most modern number theory textbooks present a few divisibility tests and explain why they work; a quick search on the Internet uncovers many articles that treat divisibility by the numbers 2 – 12, and a few that address divisibility by the primes 13, 17, and 19.
Disclaimer: Let’s be honest – these tests aren’t particularly practical in