## Euclid's Algorithm

Synonyms

## Euclidean Algorithm

(Or "Euclidean Algorithm") An algorithm for finding the greatest common divisor (GCD) of two numbers. It relies on the identity

gcd(a, b) = gcd(a-b, b)

To find the GCD of two numbers by this algorithm, repeatedly replace the larger by subtracting the smaller from it until the two numbers are equal. E.g. 132, 168 -> 132, 36 -> 96, 36 -> 60, 36 -> 24, 36 -> 24, 12 -> 12, 12 so the GCD of 132 and 168 is 12.

This algorithm requires only subtraction and comparison operations but can take a number of steps proportional to the difference between the initial numbers (e.g. gcd(1, 1001) will take 1000 steps).