Euclidean algorithms (Basic and Extended)

Difficulty Level : Medium
Last Updated : 20 Jan, 2021

GCD of two numbers is the largest number that divides both of them. A simple way to find GCD is to factorize both numbers and multiply common prime factors.

GCD

Basic Euclidean Algorithm for GCD
The algorithm is based on the below facts.

If we subtract a smaller number from a larger (we reduce a larger number), GCD doesn’t change. So if we keep subtracting repeatedly the larger of two, we end up with GCD.
Now instead of subtraction, if we divide the smaller number, the algorithm stops when we find remainder 0.

Below is a recursive function to evaluate gcd using Euclid’s algorithm.

CPP

// C++ program to demonstrate
// Basic Euclidean Algorithm
#include <bits/stdc++.h>
using namespace std;
 
// Function to return 
// gcd of a and b
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}
 
// Driver Code
int main()
{
    int a = 10, b = 15;
    cout << "GCD(" << a << ", "
         << b << ") = " << gcd(a, b) 
                        << endl;
    a = 35, b = 10;
    cout << "GCD(" << a << ", "
         << b << ") = " << gcd(a, b) 
                        << endl;
    a = 31, b = 2;
    cout << "GCD(" << a << ", "
         << b << ") = " << gcd(a, b) 
                        << endl;
    return 0;
}
 
// This code is contributed 
// by Nimit Garg

C

// C program to demonstrate Basic Euclidean Algorithm
#include <stdio.h>
 
// Function to return gcd of a and b
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b%a, a);
}
 
// Driver program to test above function
int main()
{
    int a = 10, b = 15;
    printf("GCD(%d, %d) = %dn", a, b, gcd(a, b));
    a = 35, b = 10;
    printf("GCD(%d, %d) = %dn", a, b, gcd(a, b));
    a = 31, b = 2;
    printf("GCD(%d, %d) = %dn", a, b, gcd(a, b));
    return 0;
}

Java

// Java program to demonstrate working of extended
// Euclidean Algorithm
 
import java.util.*;
import java.lang.*;
 
class GFG
{
    // extended Euclidean Algorithm
    public static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
         
        return gcd(b%a, a);
    }
 
// Driver Program
    public static void main(String[] args)
    {
        int a = 10, b = 15, g;
        g = gcd(a, b);
        System.out.println("GCD(" + a +  " , " + b+ ") = " + g);
         
        a = 35; b = 10;
        g = gcd(a, b);
        System.out.println("GCD(" + a +  " , " + b+ ") = " + g);
         
        a = 31; b = 2;
        g = gcd(a, b);
        System.out.println("GCD(" + a +  " , " + b+ ") = " + g);
 
    }
}
// Code Contributed by Mohit Gupta_OMG <(0_o)>

Python3

# Python program to demonstrate Basic Euclidean Algorithm
 
 
# Function to return gcd of a and b
def gcd(a, b): 
    if a == 0 :
        return b 
     
    return gcd(b%a, a)
 
a = 10
b = 15
print("gcd(", a , "," , b, ") = ", gcd(a, b))
 
a = 35
b = 10
print("gcd(", a , "," , b, ") = ", gcd(a, b))
 
a = 31
b = 2
print("gcd(", a , "," , b, ") = ", gcd(a, b))
 
# Code Contributed By Mohit Gupta_OMG <(0_o)>

C#

using System;
 
class GFG
{
    public static int gcd(int a, int b)
    {
        if (a == 0)
            return b;
         
        return gcd(b % a, a);
    }
     
    // Driver Code
    static public void Main ()
    {
        int a = 10, b = 15, g;
        g = gcd(a, b);
        Console.WriteLine("GCD(" + a + 
              " , " + b + ") = " + g);
         
        a = 35; b = 10;
        g = gcd(a, b);
        Console.WriteLine("GCD(" + a + 
              " , " + b + ") = " + g);
         
        a = 31; b = 2;
        g = gcd(a, b);
        Console.WriteLine("GCD(" + a + 
              " , " + b + ") = " + g);
    }
}
 
// This code is contributed by ajit

PHP

<?php
// PHP program to demonstrate
// Basic Euclidean Algorithm
 
// Function to return
// gcd of a and b
function gcd($a, $b)
{
    if ($a == 0)
        return $b;
    return gcd($b % $a, $a);
}
 
// Driver Code
$a = 10; $b = 15;
echo "GCD(",$a,"," , $b,") = ", 
                   gcd($a, $b);
echo "\n";
$a = 35; $b = 10;
echo "GCD(",$a ,",",$b,") = ", 
                  gcd($a, $b);
echo "\n";
$a = 31; $b = 2;
echo "GCD(",$a ,",", $b,") = ", 
                   gcd($a, $b);
 
// This code is contributed by m_kit
?>

Output :

GCD(10, 15) = 5
GCD(35, 10) = 5
GCD(31, 2) = 1

Time Complexity: O(Log min(a, b))
Extended Euclidean Algorithm:
Extended Euclidean algorithm also finds integer coefficients x and y such that:

  ax + by = gcd(a, b)

Examples:

Input: a = 30, b = 20
Output: gcd = 10
        x = 1, y = -1
(Note that 30*1 + 20*(-1) = 10)

Input: a = 35, b = 15
Output: gcd = 5
        x = 1, y = -2
(Note that 35*1 + 15*(-2) = 5)

The extended Euclidean algorithm updates results of gcd(a, b) using the results calculated by recursive call gcd(b%a, a). Let values of x and y calculated by the recursive call be x₁ and y₁. x and y are updated using the below expressions.

x = y₁ - ⌊b/a⌋ * x₁
y = x₁

Below is an implementation based on the above formulas.

C++

// C++ program to demonstrate working of 
// extended Euclidean Algorithm 
#include <bits/stdc++.h> 
using namespace std;
 
// Function for extended Euclidean Algorithm 
int gcdExtended(int a, int b, int *x, int *y) 
{ 
    // Base Case 
    if (a == 0) 
    { 
        *x = 0; 
        *y = 1; 
        return b; 
    } 
 
    int x1, y1; // To store results of recursive call 
    int gcd = gcdExtended(b%a, a, &x1, &y1); 
 
    // Update x and y using results of 
    // recursive call 
    *x = y1 - (b/a) * x1; 
    *y = x1; 
 
    return gcd; 
} 
 
// Driver Code
int main() 
{ 
    int x, y, a = 35, b = 15; 
    int g = gcdExtended(a, b, &x, &y); 
    cout << "GCD(" << a << ", " << b 
         << ") = " << g << endl;
    return 0; 
} 
 
// This code is contributed by TusharSabhani

C

// C program to demonstrate working of extended
// Euclidean Algorithm
#include <stdio.h>
 
// C function for extended Euclidean Algorithm
int gcdExtended(int a, int b, int *x, int *y)
{
    // Base Case
    if (a == 0)
    {
        *x = 0;
        *y = 1;
        return b;
    }
 
    int x1, y1; // To store results of recursive call
    int gcd = gcdExtended(b%a, a, &x1, &y1);
 
    // Update x and y using results of recursive
    // call
    *x = y1 - (b/a) * x1;
    *y = x1;
 
    return gcd;
}
 
// Driver Program
int main()
{
    int x, y;
    int a = 35, b = 15;
    int g = gcdExtended(a, b, &x, &y);
    printf("gcd(%d, %d) = %d", a, b, g);
    return 0;
}

Java

// Java program to demonstrate working of extended
// Euclidean Algorithm
 
import java.util.*;
import java.lang.*;
 
class GFG
{
    // extended Euclidean Algorithm
    public static int gcdExtended(int a, int b, int x, int y)
    {
        // Base Case
        if (a == 0)
        {
            x = 0;
            y = 1;
            return b;
        }
 
        int x1=1, y1=1; // To store results of recursive call
        int gcd = gcdExtended(b%a, a, x1, y1);
 
        // Update x and y using results of recursive
        // call
        x = y1 - (b/a) * x1;
        y = x1;
 
        return gcd;
    }
 
// Driver Program
    public static void main(String[] args)
    {
        int x=1, y=1;
        int a = 35, b = 15;
        int g = gcdExtended(a, b, x, y);
        System.out.print("gcd(" + a +  " , " + b+ ") = " + g);
 
    }
}
// Code Contributed by Mohit Gupta_OMG <(0-o)>

Python3

# Python program to demonstrate working of extended 
# Euclidean Algorithm 
    
# function for extended Euclidean Algorithm 
def gcdExtended(a, b): 
 
    # Base Case 
    if a == 0 :  
        return b, 0, 1
            
    gcd, x1, y1 = gcdExtended(b%a, a) 
    
    # Update x and y using results of recursive 
    # call 
    x = y1 - (b//a) * x1 
    y = x1 
    
    return gcd, x, y
     
 
# Driver code
a, b = 35,15
g, x, y = gcdExtended(a, b) 
print("gcd(", a , "," , b, ") = ", g) 

C#

// C# program to demonstrate working 
// of extended Euclidean Algorithm
using System;
 
class GFG
{
     
    // extended Euclidean Algorithm
    public static int gcdExtended(int a, int b, 
                                  int x, int y)
    {
        // Base Case
        if (a == 0)
        {
            x = 0;
            y = 1;
            return b;
        }
 
        // To store results of
        // recursive call
        int x1 = 1, y1 = 1; 
        int gcd = gcdExtended(b % a, a, x1, y1);
 
        // Update x and y using 
        // results of recursive call
        x = y1 - (b / a) * x1;
        y = x1;
 
        return gcd;
    }
     
    // Driver Code
    static public void Main ()
    {
        int x = 1, y = 1;
        int a = 35, b = 15;
        int g = gcdExtended(a, b, x, y);
        Console.WriteLine("gcd(" + a + " , " + 
                              b + ") = " + g);
    }
}
 
// This code is contributed by m_kit

PHP

<?php
// PHP program to demonstrate 
// working of extended 
// Euclidean Algorithm
 
// PHP function for 
// extended Euclidean 
// Algorithm
function gcdExtended($a, $b,    
                     $x, $y)
{
    // Base Case
    if ($a == 0)
    {
        $x = 0;
        $y = 1;
        return $b;
    }
 
    // To store results 
    // of recursive call
    $gcd = gcdExtended($b % $a, 
                       $a, $x, $y);
 
    // Update x and y using
    // results of recursive
    // call
    $x = $y - ($b / $a) * $x;
    $y = $x;
 
    return $gcd;
}
 
// Driver Code
$x = 0;
$y = 0;
$a = 35; $b = 15;
$g = gcdExtended($a, $b, $x, $y);
echo "gcd(",$a;
echo ", " , $b, ")";
echo " = " , $g;
 
// This code is contributed by ajit
?>

Output :

gcd(35, 15) = 5

How does Extended Algorithm Work?

As seen above, x and y are results for inputs a and b,
   a.x + b.y = gcd                      ----(1)  

And x₁ and y₁ are results for inputs b%a and a
   (b%a).x₁ + a.y₁ = gcd   
                    
When we put b%a = (b - (⌊b/a⌋).a) in above, 
we get following. Note that ⌊b/a⌋ is floor(b/a)

   (b - (⌊b/a⌋).a).x₁ + a.y₁  = gcd

Above equation can also be written as below
   b.x₁ + a.(y₁ - (⌊b/a⌋).x₁) = gcd      ---(2)

After comparing coefficients of 'a' and 'b' in (1) and 
(2), we get following
   x = y₁ - ⌊b/a⌋ * x₁
   y = x₁

How is Extended Algorithm Useful?
The extended Euclidean algorithm is particularly useful when a and b are coprime (or gcd is 1). Since x is the modular multiplicative inverse of “a modulo b”, and y is the modular multiplicative inverse of “b modulo a”. In particular, the computation of the modular multiplicative inverse is an essential step in RSA public-key encryption method.
References:
http://e-maxx.ru/algo/extended_euclid_algorithm
http://en.wikipedia.org/wiki/Euclidean_algorithm
http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm
This article is contributed by Ankur. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above

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