Euclidean algorithm

The Euclidean algorithm is an algorithm. It can be used to find the biggest number that divides two other numbers (the greatest common divisor of two numbers).

What the algorithm looks like in words

Euclid solved the problem graphically. He said

If you have two distances, AB and CD, and you always take away the smaller from the bigger, you will end up with a distance that measures both of them.

The algorithm as an enumerated list

Start out with two positive integers m and n.

  1. If the value of m is less than the value of n, switch the values of m and n
  2. Find a number r equal to m modulo n
  3. Let m have the same value as n
  4. Let n have the same value as r
  5. If n does not have the value of 0, go to step 2
  6. The wanted value is in m.

The algorithm in pseudocode

Note: This pseudocode uses modular arithmetic instead of subtraction. It does the same thing as above, but gets the answer faster.

Precondition: two positive integers m and n
Postcondition: the greatest common integer divisor of m and n

if m < n, swap(m,n)
while n does not equal 0
   r = m mod n
   m = n
   n = r
endwhile
output m

C/C++ source code

Iterative (Non-recursive):

int euclid_gcd(int m, int n)
{
        int temp = 0;
        if(m < n)
        {
                temp = m;
                m = n;
                n = temp;
        }
        while(n != 0)
        {
                temp = m % n;
                m = n;
                n = temp;
        }
        return m;
}

Recursive:

int euclid_gcd_recur(int m, int n)
{
        if(n == 0)
                return m;
        else
                return euclid_gcd_recur(n, m % n);
}

Images


Euclidean algorithm for Kids. Kiddle Encyclopedia.