Greatest common divisor (GCD)

For integers $a,b$ not both zero, their greatest common divisor is the largest positive integer dividing both. It is unchanged by replacing either input by its negative. The convention $\gcd (0,0)=0$ is useful, but this exceptional value is not a largest positive common divisor. For positive inputs, the Euclidean algorithm computes the answer, and Bézout's identity expresses it as an integer linear combination of the inputs.

For $12=2^2\cdot 3$ and $18=2\cdot 3^2$, a common divisor can contain at most one copy of the prime number two and one of the prime number three. Taking the smaller prime valuation at each prime number therefore givesThe check $6=18-12$ is also a Bézout's identity: every common divisor divides this difference, and six itself divides both inputs. More generally, every common divisor divides the greatest common divisor, a stronger statement than merely being no larger than it. The companion least common multiple takes the larger prime valuations, as the diagram there illustrates. For an application, divide the numerator and denominator of the fraction $12/18$ by six to obtain $2/3$; these remaining integers are coprime.