Greatest common divisor

The greatest common divisor (GCD) of two integers that are not both zero is the largest positive integer that divides both of them. It's their largest shared factor. Like the least common multiple, it can be efficiently found with prime factorizations.

For two integers \(a\) and \(b\), their greatest common divisor is denoted \(\gcd(a, b)\), or equivalently, \(\gcd(b, a).\)

For any integer \(x \neq 0\), \(\gcd(x, 0) = |x|\), which should make sense. Any positive integer divides \(0.\) The largest positive integer that divides \(x\) is its absolute value. Therefore, \(|x|\) is the largest positive integer capable of dividing both \(x\) and \(0.\)

Contents

Coprimality

Two positive integers are considered coprime (or "relatively prime") to each other if their greatest common divisor is \(1.\) Here are some notable properties that two integers will have if and only if they are coprime.

  • No prime can divide both coprime integers.
  • The least common multiple of two coprime integers is their product: \(\text{lcm}(a, b) = a \times b.\)
Logic & Proofs
Integer • Rational number • Inequality • Real number • Theorem • Proof • Statement • Proof by exhaustion • Universal generalization • Counterexample • Existence proof • Existential instantiation • Axiom • Logic • Truth • Proposition • Compound proposition • Logical operation • Logical equivalence • Tautology • Contradiction • Logic law • Predicate • Domain • Quantifier • Argument • Rule of inference • Logical proof • Direct proof • Proof by contrapositive • Irrational number • Proof by contradiction • Proof by cases • Summation • Disjunctive normal form
Set Theory
Set • Element • Empty set • Universal set • Subset • Power set • Cartesian product • String • Binary string • Empty string • Set operation • Set identity • Set proof
Functions
Function • Floor function • Ceiling function • Inverse function
Algorithms
Algorithm • Pseudocode • Command • Asymptotic notation • Time complexity • Atomic operation • Brute-force algorithm
Relations
Relation • Reflexive relation • Symmetric relation • Transitive relation • Relation composition • Equivalence relation • Equivalence class
Number Theory
Integer division • Linear combination • Division algorithm • Modular arithmetic • Prime factorization • Greatest common divisor • Least common multiple • Primality test • Factoring algorithm • Euclid's theorem • Prime number theorem • Euclidean algorithm
Induction
Proof by induction • Fibonacci sequence • Proof by strong induction • Well-ordering principle • Sequence • Factorial • Recursive definition
Combinatorics
Rule of product • Rule of sum • Bijection rule • Permutation • Combination • Complement rule • Experiment • Outcome • Sample space • Event • Probability • Probability distribution • Uniform distribution • Multiset • Sixfold way • Inclusion-exclusion principle • Pigeonhole principle
Graph Theory
Graph • Walk • Subgraph • Regular graph • Complete graph • Empty graph • Cycle graph • Hypercube graph • Bipartite graph • Component • Eulerian circuit • Eulerian trail • Hamiltonian cycle • Hamiltonian path • Tree • Huffman tree • Substring • Forest • Path graph • Star • Spanning tree • Weighted graph • Minimum spanning tree • Greedy algorithm • Prim's algorithm
Recursion
Recursion • Recursive algorithm • Correctness proof • Divide-and-conquer algorithm • Sorting algorithm • Merge sort