Euclid's Algorithm

For any non-negative integer a and any positive integer b, gcd(a, b) = gcd(b, a mod b)

Euclid(a,b) $\;\;\;\;\;$ $\;\;\;\;\;$ $\;\;\;\;\;$; second argument is strictly decreasing
1 $\;\;\;\;\;$if b = 0
2 $\;\;\;\;\;$then return a
3 $\;\;\;\;\;$else return Euclid(b, a mod b)

Example

Let a = 2322, b = 654. 
2322 = 654*3 + 360        gcd(2322,654) = gcd(654,360)
654 = 360*1 + 294         gcd(654,360) = gcd(360,294)
360 = 294*1 + 66          gcd(360,294) = gcd(294,66)
294 = 66*4 + 30           gcd(294,66) = gcd(66,30)
66 = 30*2 + 6             gcd(66,30) = gcd(30,6)
30 = 6*5                  gcd(30,6) = 6
                          gcd(6,0) = 6 (Elementary property of gcd)
Therefore, gcd(2322,654) = 6.

Euclidean Algorithm Applet