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 C function to evaluate gcd using Euclid’s algorithm.

## 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)>

**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 x_{1} and y_{1}. x and y are updated using below expressions.

x = y_{1}- ⌊b/a⌋ * x_{1}y = x_{1}

Below is C 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 = 1 y = 1 a = 35 b = 15 g = gcdExtended(a, b, x, y) print("gcd(", a , "," , b, ") = ", g) # Code Contributed by Mohit Gupta_OMG <(0_o)>

**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 x_{1}and y_{1}are results for inputs b%a and a (b%a).x_{1}+ a.y_{1}= 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).x_{1}+ a.y_{1}= gcd Above equation can also be written as below b.x_{1}+ a.(y_{1}- (⌊b/a⌋).x_{1}) = gcd ---(2) After comparing coefficients of 'a' and 'b' in (1) and (2), we get following x = y_{1}- ⌊b/a⌋ * x_{1}y = x_{1}

**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