Residue Classes

Residue classes are groups of numbers that share the same remainder when divided by a specific number, called the modulus. For example, when dividing by 3, the numbers 0, 3, 6, 9, etc., all leave a remainder of 0, forming one residue class. Similarly, 1, 4, 7, 10, etc., form another class with remainder 1, and 2, 5, 8, 11, etc., form a third class with remainder 2. These classes partition all integers into distinct sets based on their remainders, allowing mathematicians to analyze number properties systematically within modular arithmetic systems.