This paper provides a comprehensive review of integer arithmetic, a fundamental area of number theory, exploring its core concepts, historical evolution, and contemporary applications. Integer arithmetic, which involves operations with whole numbers, is foundational to advanced mathematical theories and has broad implications in fields such as cryptography, computer science, and coding theory. The study begins with a historical overview, tracing the development of integer arithmetic from ancient civilizations to modern mathematical research. It highlights significant contributions by key mathematicians like Euler, and Gauss, whose pioneering work laid the groundwork for contemporary number theory.