The Euclidean algorithm is a way to find the greatest common divisor of two positive integers. 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.
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:
// 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 code int main()
{ int a = 10, b = 15;
// Function call
printf ( "GCD(%d, %d) = %d\n" , a, b, gcd(a, b));
a = 35, b = 10;
printf ( "GCD(%d, %d) = %d\n" , a, b, gcd(a, b));
a = 31, b = 2;
printf ( "GCD(%d, %d) = %d\n" , a, b, gcd(a, b));
return 0;
} |
// 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;
// Function call
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;
} |
// Java program to demonstrate Basic Euclidean Algorithm import java.lang.*;
import java.util.*;
class GFG {
// extended Euclidean Algorithm
public static int gcd( int a, int b)
{
if (a == 0 )
return b;
return gcd(b % a, a);
}
// Driver code
public static void main(String[] args)
{
int a = 10 , b = 15 , g;
// Function call
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 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)
# Driver code if __name__ = = "__main__" :
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# program to demonstrate Basic Euclidean Algorithm 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 program to demonstrate Basic Euclidean Algorithm <?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;
// Function call 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 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 |
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 the 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.
ax + by = gcd(a, b)
gcd(a, b) = gcd(b%a, a)
gcd(b%a, a) = (b%a)x1 + ay1
ax + by = (b%a)x1 + ay1
ax + by = (b – [b/a] * a)x1 + ay1
ax + by = a(y1 – [b/a] * x1) + bx1Comparing LHS and RHS,
x = y1 – ?b/a? * x1
y = x1
Below is an implementation of the above approach:
// 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 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 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);
}
} |
# 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# 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 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 ;
?> |
<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(log N)
Auxiliary Space: O(log N)
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.