--- title: Divisibility description: This fundamental concept of arithmetic examines how integers divide one another without leaving a remainder. Learners will understand prime factorization, the Euclidean algorithm, greatest common divisors, and the foundational properties of the integers. category: mathematics subcategory: number-theory difficulty: beginner, intermediate, advanced url: /subject/divisibility --- # Divisibility This fundamental concept of arithmetic examines how integers divide one another without leaving a remainder. Learners will understand prime factorization, the Euclidean algorithm, greatest common divisors, and the foundational properties of the integers. ## Available Resources 1 Videos • 3 Books • 1 Courses • 2 Websites • 1 Papers ## Websites ### 1. Art of Problem Solving - Divisibility Community-maintained Art of Problem Solving wiki page defining divisibility of integers and collecting the standard divisibility rules for small divisors, with links to related number theory articles and competition problems useful for MATHCOUNTS and AMC preparation. **Difficulty:** Beginner | **Price:** Free **Link:** https://artofproblemsolving.com/wiki/index.php/Divisibility **Tags:** number-theory, divisibility, divisibility-rules, competition-math ### 2. 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. Elementary Number Theory Explore the beauty of numbers! This elementary number theory course covers divisibility, primes, congruences, and more. **Difficulty:** Beginner | **Price:** Free **Link:** https://www.khanacademy.org/math/number-theory **Tags:** number-theory, divisibility, modular-arithmetic, primes ## Books ### 1. Introduction to Number Theory (Art of Problem Solving) **Author:** Mathew Crawford Competition-oriented textbook moving from divisibility rules and prime factorization through GCD, LCM, base arithmetic and modular arithmetic. Each chapter teaches through worked contest problems with full solutions, aimed at MATHCOUNTS and AMC-level students. **Difficulty:** Beginner | **Language:** English | **Price:** Paid **Link:** https://www.amazon.com/dp/1934124125?tag=edmonddante07-20 **Tags:** number-theory, competition-math, modular-arithmetic, prime-factorization, divisibility ### 2. Elementary Number Theory: Primes, Congruences, and Secrets **Author:** William Stein Springer Undergraduate Texts volume the author released free online with the publisher's permission. Treats primes, the Euclidean algorithm, unique factorization, congruences and quadratic reciprocity, then RSA and elliptic curves, with every computation worked in open-source Sage. Answers 'why does divisibility matter' by carrying factorization and gcd straight into RSA and elliptic curves, with runnable computations. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://wstein.org/ent/ **Tags:** number-theory, congruences, cryptography, elliptic-curves, sagemath ### 3. Elementary Number Theory (Raji) **Author:** Wissam Raji Open-access undergraduate textbook by Wissam Raji of the American University of Beirut. Opens with the division algorithm, greatest common divisors, the Euclidean algorithm and Bezout's identity, then builds to primes, unique factorization, congruences and multiplicative functions. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Elementary_Number_Theory_(Raji) **Tags:** number-theory, divisibility, euclidean-algorithm, congruences, prime-factorization ## Videos ### 1. Lecture 4: Number Theory I (MIT 6.042J) **Author:** Tom Leighton, Marten van Dijk Covers divisibility, greatest common divisors as linear combinations, Euclid's algorithm and the Pulverizer (extended Euclidean algorithm) for finding Bezout coefficients. A full lecture video from a 25-video discrete mathematics course with exams and solutions; afterwards you can run both algorithms by hand. **Difficulty:** Beginner | **Language:** English | **Price:** Free **Link:** https://ocw.mit.edu/courses/6-042j-mathematics-for-computer-science-fall-2010/resources/lecture-4-number-theory-i/ **Tags:** divisibility, greatest-common-divisor, euclidean-algorithm, bezout-identity, number-theory ## Papers ### 1. Divisibility and Greatest Common Divisors (Keith Conrad) **Author:** Keith Conrad Expository note from a UConn number theorist. Establishes the divisibility relation and its properties, then proves the two central theorems on greatest common divisors — Euclid's algorithm and Bezout's identity — with worked integer examples and careful notation. **Difficulty:** Intermediate | **Language:** English | **Price:** Free **Link:** https://kconrad.math.uconn.edu/blurbs/ugradnumthy/divgcd.pdf **Tags:** number-theory, divisibility, greatest-common-divisor, euclidean-algorithm, bezout-identity --- *This content is part of Dantes.io - Your Treasure Map to Knowledge* *Curated by humans at Dantes.io. Personal study use welcome; republishing this curation requires permission (team@dantes.io).* View this page online: https://dantes.io/subject/divisibility