Dave4Math / Courses / Divisibility in the Integers

Divisibility in the Integers

Master divisibility in the integers. Learn primes, Euclidean Algorithm, Fundamental Theorem of Arithmetic, and linear Diophantine equations. From numbers to number theory, from examples to proofs, this course helps you make these transitions. 


January 11, 2021

Created by Dave

This course begins Summer 2021.
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For many math students learning a new topic can be uneasy and sometimes even cause apprehension. But often times, suddenly out of nowhere, is that feeling that you got it. You understand! Introducing Dave’s online math course Divisibility in the Integers. You’ll be glad it was him that taught it, because you’ll get that feeling over and over again. 

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in divisibility 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 divisibility, primes, greatest common divisors, Euclidean algorithm, fundamental theorem of algebra, and linear Diophantine equations.


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 Divisibility in the Integers

  • Definition of Divisibility
  • Divisibility Lemmas
  • The Division Algorithm
  • Examples Using the Division Algorithm
  • Sieve of Eratosthenes
  • Prime Divisors
  • Infinitude of Primes
  • Definition of Greatest Common Divisors
  • Bezout’s Identity
  • Relatively Prime
  • Euclidean Algorithm Lemma
  • Euclid’s Algorithm
  • The Fundamental Theorem of Arithmetic
  • Characterization of Primes
  • Proof of the Fundamental Theorem
  • Least Common Multiples
  • What are Diophantine Equations?
  • Two Variable Linear Diophantine Equations
  • Solving Linear Diophantine Equations
  • Multivariable Linear Diophantine Equations

Course Description

This course is a continuation of the course entitled The Natural Numbers. You are required to know mathematical induction for this course.

This course begins with the definition of divisibility in the integers and then discusses the division algorithm in great detail. We then discuss the prime numbers (the fundamental building blocks of the integers).

Then we examine the greatest common divisors of two integers and how to find them using the Euclidean algorithm. After this, we state and prove the fundamental theorem of arithmetic. Towards the end of the course, we include linear Diophantine equations. We emphasize solvability and solve many examples.

I recommend the prerequisite course The Natural Numbers. 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.