Skip to content
Related Articles

Related Articles

Euclidean algorithms (Basic and Extended)

View Discussion
Improve Article
Save Article
  • Difficulty Level : Medium
  • Last Updated : 21 Jul, 2022

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 one (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 the 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;
}

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
?>
Javascript



<script>
 
// JavaScript 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
 
    let a = 10, b = 15;
   document.write( "GCD(" + a + ", "
         + b + ") = " + gcd(a, b) +"<br/>");
          
    a = 35, b = 10;
   document.write( "GCD(" + a + ", "
         + b + ") = " + gcd(a, b) +"<br/>");
          
    a = 31, b = 2;
    document.write( "GCD(" + a + ", "
         + b + ") = " + gcd(a, b) +"<br/>");
 
// This code contributed by aashish1995
 
</script>
Output : 
GCD(10, 15) = 5
GCD(35, 10) = 5
GCD(31, 2) = 1
Time Complexity: O(Log min(a, b)), Auxiliary Space: 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 the 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
Recommended Practice
Required Rooms
Try It!
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;
}
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.lang.*;
import java.util.*;
 
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);
    }
}
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);
    }
}
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 = $y - floor($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;
 
?>
Javascript



<script>
 
// Javascript program to demonstrate
// working of extended
// Euclidean Algorithm
 
// Javascript 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
    let 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
let x = 0;
let y = 0;
let a = 35;
let b = 15;
let g = gcdExtended(a, b, x, y);
document.write("gcd(" + a);
document.write(", " + b + ")");
document.write(" = " + g);
 
 
</script>
Output : 
gcd(35, 15) = 5
Time Complexity: O(logn), Auxiliary Space: O(logn)
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.
This article is contributed by Ankur. Please write comments if you find anything incorrect, or if you want to share more information about the topic discussed above

My Personal Notes
arrow_drop_up
Previous
Pascal's Triangle
Next

Find x and y satisfying ax + by = n
Recommended Articles
Page :
C Program for Basic Euclidean algorithms
29, May 15
Java Program for Basic Euclidean algorithms
29, May 15
Pairs with same Manhattan and Euclidean distance
02, Jan 19
Find sum of Kth largest Euclidean distance after removing ith coordinate one at a time
15, Sep 21
Time Complexity of Euclidean Algorithm
24, Mar 21
C program to find the Euclidean distance between two points
04, Nov 21
Minimum Sum of Euclidean Distances to all given Points
12, Aug 20
Calculate the Square of Euclidean Distance Traveled based on given conditions
19, Aug 20
Find HCF of two numbers without using recursion or Euclidean algorithm
12, Apr 19
Extended Midy's theorem
02, Mar 18
Basic Operators in Java
20, Feb 17
Menu driven program in C++ to perform various basic operations on array
02, Feb 21
Minimum number of basic logic gates required to realize given Boolean expression
15, Mar 21
Number Theory (Interesting Facts and Algorithms)
02, Jul 17
Custom Building Cryptography Algorithms (Hybrid Cryptography)
06, Sep 18
Classification of Algorithms with Examples
05, Jul 21
Max count of N using digits of M such that 2 and 5, and, 6 and 9 can be treated as same respectively
13, Jul 21
Count ways to generate pairs having Bitwise XOR and Bitwise AND equal to X and Y respectively
16, Sep 20
Numbers of Length N having digits A and B and whose sum of digits contain only digits A and B
31, Jan 19
Count of Ways to Choose N People With at Least X Men and Y Women from P Men and Q Women | Set 2
30, Nov 21
Reach A and B by multiplying them with K and K^2 at every step
26, Mar 18
Sum of elements in range L-R where first half and second half is filled with odd and even numbers
08, Aug 18
Sum and product of k smallest and k largest prime numbers in the array
15, Nov 18
Count pairs (A, B) such that A has X and B has Y number of set bits and A+B = C
27, Nov 18
Article Contributed By :
https://media.geeksforgeeks.org/auth/avatar.png
GeeksforGeeks
Vote for difficulty
Current difficulty :
Medium





Improved By :
Article Tags :
Practice Tags :
Report Issue

We use cookies to ensure you have the best browsing experience on our website. By using our site, you
acknowledge that you have read and understood our
Cookie Policy &
        Privacy Policy

Lightbox

Start Your Coding Journey Now!