---
title: Congruences
description: This area of number theory studies the equivalence of integers relative to a given modulus. Learners will understand how to solve linear and quadratic congruences, apply modular arithmetic, and use key results like Fermat's Little Theorem.
category: mathematics
subcategory: number-theory
difficulty: beginner, intermediate, advanced
url: /subject/congruences
---

# Congruences

This area of number theory studies the equivalence of integers relative to a given modulus. Learners will understand how to solve linear and quadratic congruences, apply modular arithmetic, and use key results like Fermat's Little Theorem.

## Where to start

Start with Modular Arithmetic - Khan Academy, free interactive lessons on congruences. 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. For more on congruences and their applications, the AoPS Community's Introduction to Number Theory course follows naturally.

## Available Resources

2 Books • 1 Courses • 3 Websites

## Websites

### 1. Modular Arithmetic - Khan Academy

Interactive lessons on congruences

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

**Link:** https://www.khanacademy.org/computing/computer-science/cryptography/modarithmetic

### 2. Number Theory: In Context and Interactive

**Author:** Karl-Dieter Crisman

Free online undergraduate textbook with embedded SageMath cells. It has dedicated chapters on first steps with congruence, linear congruences, general congruences, the group of units and Euler's function, primitive roots, solving quadratic congruences and quadratic reciprocity. The interactive Sage cells make the computations concrete for beginners.

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

**Link:** https://math.gordon.edu/ntic/

### 3. 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

## Books

### 1. A Classical Introduction to Modern Number Theory (2nd Edition)

**Author:** Kenneth Ireland, Michael Rosen

The canonical next step once a reader knows the elementary theory: Springer Graduate Texts in Mathematics 84. Early chapters treat congruences, the structure of the unit group mod n and quadratic reciprocity, with proofs via Gauss sums. Later chapters go on to cubic and biquadratic reciprocity, finite fields and zeta functions.

**Difficulty:** Advanced | **Language:** English | **Price:** Paid

**Link:** https://www.amazon.com/dp/038797329X?tag=edmonddante07-20

### 2. An Introduction to the Theory of Numbers (5th Edition)

**Author:** Ivan Niven, Herbert S. Zuckerman, Hugh L. Montgomery

Widely regarded as the standard reference for elementary congruence theory, with large, carefully graded exercise sets. Standard undergraduate number theory text, first published in 1960. Chapter 2 develops congruences, including solutions of congruences, CRT, prime-power moduli, primitive roots and power residues. Chapter 3 covers quadratic residues, the Legendre and Jacobi symbols and quadratic reciprocity.

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

**Link:** https://www.amazon.com/dp/0471625469?tag=edmonddante07-20

## 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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