Euclidean algorithms (Basic and Extended)
Last Updated : 17 Feb, 2025
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.
Examples:
input: a = 12, b = 20
Output: 4
Explanation: The Common factors of (12, 20) are 1, 2, and 4 and greatest is 4.
input: a = 18, b = 33
Output: 3
Explanation: The Common factors of (18, 33) are 1 and 3 and greatest is 3.
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 larger number, the algorithm stops when we find the remainder 0.
CPP // C++ program to demonstrate working of // extended Euclidean Algorithm #include <bits/stdc++.h> using namespace std; // Function to return // gcd of a and b int findGCD(int a, int b) { if (a == 0) return b; return findGCD(b % a, a); } int main() { int a = 35, b = 15; int g = findGCD(a, b); cout << g << endl; return 0; } // C program to demonstrate working of // extended Euclidean Algorithm #include <stdio.h> // Function to return // gcd of a and b int findGCD(int a, int b) { if (a == 0) return b; return findGCD(b % a, a); } int main() { int a = 35, b = 15; int g = findGCD(a, b); printf("%d\n", g); return 0; } // Java program to demonstrate working of // extended Euclidean Algorithm class GFG { // Function to return // gcd of a and b static int findGCD(int a, int b) { if (a == 0) return b; return findGCD(b % a, a); } public static void main(String[] args) { int a = 35, b = 15; int g = findGCD(a, b); System.out.println(g); } } # Python program to demonstrate working of # extended Euclidean Algorithm # Function to return # gcd of a and b def findGCD(a, b): if a == 0: return b return findGCD(b % a, a) # Main function def main(): a, b = 35, 15 g = findGCD(a, b) print(g) if __name__ == "__main__": main()
// C# program to demonstrate working of // extended Euclidean Algorithm using System; class GFG { // Function to return // gcd of a and b static int FindGCD(int a, int b) { if (a == 0) return b; return FindGCD(b % a, a); } public static void Main() { int a = 35, b = 15; int g = FindGCD(a, b); Console.WriteLine(g); } } // JavaScript program to demonstrate working of // extended Euclidean Algorithm // Function to return // gcd of a and b function findGCD(a, b) { if (a === 0) return b; return findGCD(b % a, a); } function main() { let a = 35, b = 15; let g = findGCD(a, b); console.log(g); } // Run the main function main(); OutputGCD(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
Explanation: 30*1 + 20*(-1) = 10
Input: a = 35, b = 15
Output: gcd = 5, x = 1, y = -2
Explanation: 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) + bx1
Comparing LHS and RHS,
x = y1 - \lfloor b/a \rfloor* x1
y = x1
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; 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; } int findGCD(int a, int b) { int x = 1, y = 1; return gcdExtended(a, b, x, y); } int main() { int a = 35, b = 15; int g = findGCD(a, b); cout << g << endl; return 0; } // C program to demonstrate working of // extended Euclidean Algorithm #include <stdio.h> // 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; 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; } int findGCD(int a, int b) { int x = 1, y = 1; return gcdExtended(a, b, &x, &y); } int main() { int a = 35, b = 15; int g = findGCD(a, b); printf("%d\n", g); return 0; } // Java program to demonstrate working of // extended Euclidean Algorithm class GFG { // Function for extended Euclidean Algorithm static int gcdExtended(int a, int b, int[] x, int[] y) { // Base Case if (a == 0) { x[0] = 0; y[0] = 1; return b; } int[] x1 = {0}, y1 = {0}; int gcd = gcdExtended(b % a, a, x1, y1); // Update x and y using results of // recursive call x[0] = y1[0] - (b / a) * x1[0]; y[0] = x1[0]; return gcd; } static int findGCD(int a, int b) { int[] x = {1}, y = {1}; return gcdExtended(a, b, x, y); } public static void main(String[] args) { int a = 35, b = 15; int g = findGCD(a, b); System.out.println(g); } } # 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] = 0 y[0] = 1 return b x1, y1 = [0], [0] gcd = gcdExtended(b % a, a, x1, y1) # Update x and y using results of # recursive call x[0] = y1[0] - (b // a) * x1[0] y[0] = x1[0] return gcd def findGCD(a, b): x, y = [1], [1] return gcdExtended(a, b, x, y) # Main function def main(): a, b = 35, 15 g = findGCD(a, b) print(g) if __name__ == "__main__": main()
// C# program to demonstrate working of // extended Euclidean Algorithm using System; class GFG { // Function for extended Euclidean Algorithm static int GcdExtended(int a, int b, ref int x, ref int y) { // Base Case if (a == 0) { x = 0; y = 1; return b; } int x1 = 0, y1 = 0; int gcd = GcdExtended(b % a, a, ref x1, ref y1); // Update x and y using results of // recursive call x = y1 - (b / a) * x1; y = x1; return gcd; } static int FindGCD(int a, int b) { int x = 1, y = 1; return GcdExtended(a, b, ref x, ref y); } public static void Main() { int a = 35, b = 15; int g = FindGCD(a, b); Console.WriteLine(g); } } // JavaScript program to demonstrate working of // extended Euclidean Algorithm // Function for extended Euclidean Algorithm function gcdExtended(a, b, x, y) { // Base Case if (a === 0) { x[0] = 0; y[0] = 1; return b; } let x1 = [0], y1 = [0]; let gcd = gcdExtended(b % a, a, x1, y1); // Update x and y using results of // recursive call x[0] = y1[0] - Math.floor(b / a) * x1[0]; y[0] = x1[0]; return gcd; } function findGCD(a, b) { let x = [1], y = [1]; return gcdExtended(a, b, x, y); } // Main function function main() { let a = 35, b = 15; let g = findGCD(a, b); console.log(g); } // Run the main function main(); Time Complexity: O(log min(a, b))
Auxiliary Space: O(log (min(a,b)))
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 - (\lfloor b/a \rfloor).a) in above,
we get following. Note that \lfloor b/a\rfloor is floor(b/a)
(b - (\lfloor b/a \rfloor).a).x1 + a.y1 = gcd
Above equation can also be written as below
b.x1 + a.(y1 - (\lfloor b/a \rfloor).x1) = gcd ---(2)
After comparing coefficients of 'a' and 'b' in (1) and
(2), we get following,
x = y1 - \lfloor b/a \rfloor * 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.
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