Divisibility Tests

Introduction to Divisibility Tests

• Divisibility tests are simple methods to determine whether one number is divisible by another, without having to perform the division.
• These tests help in quickly identifying factors of a number, and are useful tools in number theory and in solving mathematical problems involving divisions or factors.

Common Divisibility Tests

• The test for divisibility by 2 is that the last digit of the number is even (0, 2, 4, 6, or 8).
• The test for divisibility by 3 is that the sum of the digits of the number is divisible by 3.
• The test for divisibility by 4 is that the number formed by the last two digits is divisible by 4.
• The test for divisibility by 5 is that the last digit of the number is either 0 or 5.
• The test for divisibility by 6 is that the number is divisible by both 2 and 3.
• The test for divisibility by 8 is that the number formed by the last three digits is divisible by 8.
• The test for divisibility by 9 is that the sum of the digits of the number is divisible by 9.
• The test for divisibility by 10 is that the last digit of the number is 0.
• Remember, these rules only work for the above divisors in the decimal number system.

Divisibility Tests and Prime Numbers

• Divisibility tests are often used to check the primality of a number, i.e., whether a number is prime.
• Familiarity with divisibility tests for small numbers (2, 3, 5, 7, 11) helps in the sieve of Eratosthenes, a simple, ancient algorithm for finding all prime numbers up to a given limit.

Applications of Divisibility Tests

• Divisibility tests are not merely academic tools, they are widely used in everyday life, notably in the generation and validation of Identifiers such as ISBN codes for books and UPC codes for products.
• Knowing these tests can significantly speed up mathematical calculations and provide shortcuts to solving complex problems.

Divisibility, Congruences, and Remainders

• Studying divisibility naturally leads to the topics of congruences and remainders, which are key part of Number Theory.
• Familiarise with Modular Arithmetic, which forms the mathematical foundation of divisibility tests and provides further techniques for simplifying problems involving division and remainders.

Practice Problems

• Applying the tests in practice problems is necessary to consolidate knowledge and improve speed. Practice problems of varying complexity, including those that require the application of multiple tests.
• Understanding why each divisibility test works is equally important for a full understanding. This typically involves a bit of elementary number theory and a dash of algebra.