---
title: Modular Arithmetic
description: This system of arithmetic deals with integers that wrap around a fixed value called a modulus. Learners will understand how to perform modular operations, solve systems of congruences, and apply these concepts to cryptography and computer science.
category: mathematics
subcategory: number-theory
difficulty: beginner, intermediate, advanced
url: /subject/modular-arithmetic
---

# Modular Arithmetic

This system of arithmetic deals with integers that wrap around a fixed value called a modulus. Learners will understand how to perform modular operations, solve systems of congruences, and apply these concepts to cryptography and computer science.

## Where to start

Start with the Art of Problem Solving Wiki - Modular Arithmetic, a free wiki with problems and solutions. If you only use one resource, make it the Number Theory course from UC San Diego on Coursera, which builds from division with remainder and modular arithmetic to Euclid's algorithm and RSA.

## Available Resources

1 Courses • 4 Websites

## Websites

### 1. Wikipedia - Modular Arithmetic

Encyclopedia article covering congruence relations, residue classes, modular inverses, and the ring of integers modulo n, with links to Fermat's little theorem and the Chinese remainder theorem. Useful as a reference for checking definitions, notation, and applications in cryptography and computing.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://en.wikipedia.org/wiki/Modular_arithmetic

**Tags:** modular-arithmetic, congruences, number-theory, residue-classes

### 2. Brilliant.org - Modular Arithmetic

Brilliant's community wiki page on modular arithmetic, explaining congruence notation, addition and multiplication rules, and modular exponentiation through worked examples. Readers practice reducing large expressions mod n and solving short competition-style problems on remainders and divisibility.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://brilliant.org/wiki/modular-arithmetic/

**Tags:** modular-arithmetic, congruences, number-theory, problem-solving

### 3. Art of Problem Solving Wiki - Modular Arithmetic

Art of Problem Solving's wiki entry on modular arithmetic, written for math-competition students. It defines congruences, shows how to compute with them, and works through AMC and AIME-style problems, so learners can use remainders to solve divisibility and digit problems.

**Difficulty:** Beginner | **Price:** Free

**Link:** https://artofproblemsolving.com/wiki/index.php/Modular_arithmetic

**Tags:** modular-arithmetic, congruences, number-theory, math-competitions

### 4. Wolfram MathWorld

**Author:** Eric W. Weisstein

MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://mathworld.wolfram.com

**Tags:** mathematics-reference, encyclopedia, abstract-algebra, number-theory, geometry

## Courses

### 1. Number Theory

Four-week UC San Diego course building from division with remainder and modular arithmetic to Euclid's algorithm and RSA. Finishing it, you can encrypt and decrypt messages and break RSA implementations that reuse or mishandle keys.

**Difficulty:** Intermediate | **Language:** English | **Price:** Free

**Link:** https://www.coursera.org/learn/number-theory-cryptography

**Tags:** courses, mathematics-statistics, mathematics

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