modulo2

Modulo2 refers to the remainder after division by 2, a foundational concept in modular arithmetic. For any integer a, a mod 2 is the remainder r in {0,1} such that a = 2q + r for some integer q. Equivalently, a mod 2 determines the parity of a: 0 if a is even and 1 if a is odd. In the language of congruences, a ≡ r (mod 2). The two residue classes modulo 2 are denoted [0] for even numbers and [1] for odd numbers.

The arithmetic of integers modulo 2 forms the two-element field GF(2) when equipped with addition and multiplication modulo 2. The elements are 0 and 1, with 1 + 1 = 0 and 1 × 1 = 1. In this system, addition corresponds to bitwise XOR and multiplication corresponds to bitwise AND, which makes GF(2) a common setting in digital logic and coding theory.

In computing and digital systems, modulo 2 is often interpreted as the least-significant bit of a binary number. For example, 7 mod 2 equals 1 and 8 mod 2 equals 0. This simple operation underpins parity checks and various binary algorithms. It also underlies many concepts in computer science, including error detection, linear codes, and cryptographic schemes that rely on arithmetic in GF(2).