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Understanding Modular Arithmetic: A Comprehensive Overview

Modular arithmetic is a fundamental concept in mathematics, particularly useful for understanding systems where numbers "wrap around" after reaching a certain point. Its applications are widespread, from designing digital circuits to cryptography, but perhaps the most familiar example to many is the 12-hour clock.

The Basics of Modular Arithmetic

At its core, modular arithmetic involves dividing integers by a fixed number called the modulus and considering only the remainder. For any integers (a) and (b), we say:

[ a \equiv b \pmod{n} ]

meaning (a) and (b) leave the same remainder when divided by (n). The notation (a \equiv b \pmod{n}) signifies that (a) and (b) are congruent modulo (n).