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 factors.
Basic Euclidean Algorithm for GCD
The algorithm is based on below facts.
- If we subtract smaller number from larger (we reduce 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 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 = 10b = 15print("gcd(", a , "," , b, ") = ", gcd(a, b)) a = 35b = 10print("gcd(", a , "," , b, ") = ", gcd(a, b)) a = 31b = 2print("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 "
"; $a = 35; $b = 10; echo "GCD(",$a ,",",$b,") = ", gcd($a, $b); echo "
"; $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 10*0 + 5*1 = 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 below expressions.
x = y1 - ⌊b/a⌋ * x1 y = x1
Below is implementation based on above formulas.
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, x, y): # Base Case if a == 0 : x = 0 y = 1 return b x1 = 1 y1 = 1 # To store results of recursive call 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 x = 1y = 1a = 35b = 15g = gcdExtended(a, b, x, y) print("gcd(", a , "," , b, ") = ", g) # Code Contributed by Mohit Gupta_OMG <(0_o)> |
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(a/b)
(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
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