# Finite (modular) arithmetics

## Finite (modular) arithmetics

# 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.