Greatest common divisor

Definition

Let $a$ and $b$ be integers that are not both $0$. A common divisor of $a$ and $b$ is an integer $d$ such that $d \mid a$ and $d \mid b$. The greatest common divisor $\gcd(a, b)$ is the largest common divisor of $a$ and $b$. By convention, $\gcd(0, 0) = 0$.

Remarks

The number $\gcd(a, b)$ is positive when $a$ and $b$ are not both $0$, because $1$ is a common divisor. An equivalent definition: $\gcd(a, b)$ is the unique nonnegative common divisor of $a$ and $b$ that each common divisor divides. This form also gives $\gcd(0, 0) = 0$. For all integers $a$, $\gcd(a, 0) = |a|$. The Euclidean algorithm computes $\gcd(a, b)$.

Explanations

L1 human

List the positive divisors of each number, and take the largest number that is in both lists. The divisors of $12$ are $1, 2, 3, 4, 6, 12$, and the divisors of $18$ are $1, 2, 3, 6, 9, 18$. Thus $\gcd(12, 18) = 6$.