Dave4Math / Courses / Congruence in the Integers

Congruence in the Integers

Master congruence in the integers. Learn about linear, quadratic, polynomial congruence equations, Euler's Theorem, and applications. With this course, you'll completely understand the Law of Quadratic Reciprocity (both how to apply it and how to prove it) and also the Tonelli-Shanks algorithm.


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Imagine a teacher that teaches the lessons in such a way that students who have never taken the subject before can identify with it. Introducing Dave’s online math course: Congruence in the Integers. Also, imagine that he is funny and that you enjoyed and learned a lot from each topic. Now stop imagining because it is real!   

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in congruence in the integers.
  • Students who are enrolled in Number Theory and want to improve their grade.
  • Anyone interested in majoring in mathematics, physics, or engineering.
  • Anyone wanting to learn about linear congruence equations, Chinese Remainder Theorem, solving polynomial congruence equations, and Euler’s Theorem.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A calculator is not necessary.

What You’ll Learn in Congruence in the Integers

  • Definition and Examples of Congruences
  • Congruence Lemmas
  • Least Positive Residues
  • Modular Arithmetic
  • Solving Linear Congruence Equations
  • The Inverse of a Integer Modulo n
  • Linear System of Congruence Equations
  • Chinese Remainder Theorem I
  • Chinese Remainder Theorem II
  • Congruence Reduction
  • Simple Examples of Polynomial Congruence
  • Hensel’s Lifting Theorem
  • Solving Polynomial Congruences
  • Divisibility Tests
  • Days of the Week
  • Wilson’s Theorem
  • Fermat’s Little Theorem
  • Euler’s Totient Function
  • Solving Euler Functional Equations
  • Reduced Residue Systems
  • Euler’s Theorem and Its Proof
  • General Quadratic Congruence
  • Quadratic Residues
  • The Legendre Symbol
  • Quadratic Characters
  • Properties of the Legendre Symbol
  • The Law of Quadratic Reciprocity
  • Euler’s and Gauss’s Criterion
  • A Proof of the Law of Quadratic Reciprocity
  • The Tonelli–Shanks Theorem
  • Examples of Tonelli–Shanks Algorithm
  • Solving Quadratic Congruences

Course Description

Recommended Prerequisites for Congruence in the Integers

I recommend the prerequisite course Divisibility in the Integers. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free number theory articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.