# Finite (Modular) Arithmetic

## Introduction to Finite (Modular) Arithmetic

• Finite (Modular) Arithmetic is a system of arithmetic for integers, which considers only the remainder when a number is divided by a positive integer called the modulus.
• The integer in concern in the modulus is sometimes referred to as being ‘taken mod’ the modulus.
• It is called “modular” because it involves a modulus, and “finite” because there are a finite number of possible outcomes for any given calculation.

## Understanding Modulo Operations

• To perform a modulo operation, find the remainder after division.
• “a mod n” is the remainder when a is divided by n. For instance, `9 mod 5 = 4` because 4 is the remainder when 9 is divided by 5.
• The modulo operation is often represented using the symbol `%`. For instance, in many programming languages, `9%5` will yield `4`.

## Properties of Finite (Modular) Arithmetic

• Finite (modular) arithmetic exhibits many of the same properties as regular arithmetic, including commutativity, associativity, and distributivity.
• These properties mean that, in many cases, calculations in modular arithmetic can be done in any order without changing the result, just like in normal arithmetic.
• Nevertheless, remember that division is not always possible in finite (modular) arithmetic.

## Equivalence Classes and Congruences

• In finite (modular) arithmetic, a set of integers that give the same remainder when divided by the modulus is termed an equivalence class.
• Two numbers are said to be congruent modulo n if they’re in the same equivalence class for modulus n. It’s represented as ‘ ≡ ‘.
• For Instance, 7 and 2 are congruent modulo 5, as for both the remainder is 2 when divided by 5. This will be written as `7 ≡ 2 (mod 5)`

## Finite (Modular) Arithmetic and Number Theory

• Modular arithmetic is a fundamental concept in number theory, and is used in several areas, including cryptography, computer science, and coding theory.
• In particular, it plays a key role in public key cryptography algorithms such as RSA.
• It also has applications in areas like music theory and visual art, where it can be used to create mathematical patterns.

## Problems and Exercises

• Apply the concepts and principles of modular arithmetic in various problems and exercises. This will help to consolidate understanding of the topic.
• Carry out calculations using modulo arithmetic and interpret equivalence classes and congruencies to build fluency with the concepts.
• Working with finite (modular) arithmetic problems will require both computational skills and theoretical understanding encompassing their relationship with number theory.