Residue Class
Residue Classes in Number Theory Residue Classes in Number Theory Residue classes are a fundamental concept in modular arithmetic, which is widely used in number theory, cryptography, and coding theory. This article introduces the idea of residue classes and explains their properties. What is a Residue Class? In modular arithmetic, two integers \( a \) and \( b \) are said to be congruent modulo \( m \) if they leave the same remainder when divided by \( m \). This is written as: \( a \equiv b \pmod{m} \) The number \( m \) is called the modulus, and all numbers congruent to \( a \) modulo \( m \) belong to the same **residue class**. Definition of Residue Classes Given an integer \( a \) and a modulus \( m \), the **residue class** of \( a \) modulo \( m \) is the set of all integers that are congruent to \( a \) modulo \( m \). Mathematically, it is expresse...