Program to Find GCD or HCF of Two Numbers

Given two numbers a and b, the task is to find the GCD of the two numbers.

Note: GCD (Greatest Common Divisor) or HCF (Highest Common Factor) of two numbers is the largest number that divides both of them. 

Examples:

Input: a = 20, b = 28
Output: 4
Explanation: The factors of 20 are 1, 2, 4, 5, 10 and 20. The factors of 28 are 1, 2, 4, 7, 14 and 28. Among these factors, 1, 2 and 4 are the common factors of both 20 and 28. The greatest among the common factors is 4.

Input: a = 60, b = 36
Output: 12

Naive Approach for GCD of two numbers:

The basic idea is to find the minimum of the two numbers and find its highest factor which is also a factor of the other number.

Below is the code implementation of the above idea:

C++

// C++ program to find GCD of two numbers\

#include <bits/stdc++.h>
using namespace std;

// Function to return gcd of a and b
int gcd(int a, int b)
{
    // Find Minimum of a and b
    int result = min(a, b);
    while (result > 0) {
        if (a % result == 0 && b % result == 0) {
            break;
        }
        result--;
    }

    // Return gcd of a and b
    return result;
}

// Driver code
int main()
{
    int a = 98, b = 56;
    cout << "GCD of " << a << " and " << b << " is "
         << gcd(a, b);
    return 0;
}
// This code is contributed by Suruchi Kumari

C

// C program to find GCD of two numbers

#include <math.h>
#include <stdio.h>

// Function to return gcd of a and b
int gcd(int a, int b)
{
    // Find Minimum of a and b
    int result = ((a < b) ? a : b);
    while (result > 0) {
        if (a % result == 0 && b % result == 0) {
            break;
        }
        result--;
    }

    // Return gcd of a and b
    return result;
}

// Driver code
int main()
{
    int a = 98, b = 56;
    printf("GCD of %d and %d is %d ", a, b, gcd(a, b));
    return 0;
}
// This code is contributed by Suruchi Kumari

Java

// Java program to find GCD of two numbers
import java.io.*;

public class GFG {

    // Function to return gcd of a and b
    static int gcd(int a, int b)
    {
        // Find Minimum of a and b
        int result = Math.min(a, b);
        while (result > 0) {
            if (a % result == 0 && b % result == 0) {
                break;
            }
            result--;
        }

        // Return gcd of a and b
        return result;
    }

    // Driver code
    public static void main(String[] args)
    {
        int a = 98, b = 56;
        System.out.print("GCD of " + a + " and " + b
                         + " is " + gcd(a, b));
    }
}

// This code is contributed by AnkThon

Python3

# Python program to find GCD of two numbers


# Function to find gcd of two numbers
def gcd(a, b):

    # Find minimum of a and b
    result = min(a, b)

    while result:
        if a % result == 0 and b % result == 0:
            break
        result -= 1

    # Return the gcd of a and b
    return result


# Driver Code
if __name__ == '__main__':
    a = 98
    b = 56
    print(f"GCD of {a} and {b} is {gcd(a, b)}")


# This code is contributed by Soham Mirikar

C#

// C# program to find GCD of two numbers

using System;
public class GFG {

    // Function to return gcd of a and b
    static int gcd(int a, int b)
    {
        // Find Minimum of a and b
        int result = Math.Min(a, b);
        while (result > 0) {
            if (a % result == 0 && b % result == 0) {
                break;
            }
            result--;
        }

        // Return gcd of a and b
        return result;
    }

    // Driver code
    public static void Main(string[] args)
    {
        int a = 98, b = 56;
        Console.WriteLine("GCD of " + a + " and " + b
                          + " is " + gcd(a, b));
    }
}

// This code is contributed by AnkThon

Javascript

// Javascript program to find GCD of two numbers
// Function to return gcd of a and b
function gcd(a,b)
{
    // Find Minimum of a and b
    let result = Math.min(a, b);
    while (result > 0) {
        if (a % result == 0 && b % result == 0) {
            break;
        }
        result--;
    }

    // Return gcd of a and b
    return result;
}

// Driver program to test above function
let a = 98;
let b = 56;
console.log("GCD of ",a," and ",b," is ",gcd(a, b));
// This code is contributed by akashish__

Output

GCD of 98 and 56 is 14

Time Complexity: O(min(a,b)) 
Auxiliary Space: O(1)

Euclidean algorithm for GCD of two numbers:

The idea of this algorithm is, the GCD of two numbers doesn’t change if the smaller number is subtracted from the bigger number. This is the Euclidean algorithm by subtraction. It is a process of repeat subtraction, carrying the result forward each time until the result is equal to any one number being subtracted.

Pseudo-code:

gcd(a, b):
    if a = b:
        return a
    if a > b:
        return gcd(a – b, b)
    else:
        return gcd(a, b – a)

Below is the implementation of the above approach.

C++

// C++ program to find GCD of two numbers

#include <bits/stdc++.h>
using namespace std;

// Recursive function to return gcd of a and b
int gcd(int a, int b)
{
    // Everything divides 0
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    // Base case
    if (a == b)
        return a;

    // a is greater
    if (a > b)
        return gcd(a - b, b);
    return gcd(a, b - a);
}

// Driver code
int main()
{
    int a = 98, b = 56;
    cout << "GCD of " << a << " and " << b << " is "
         << gcd(a, b);
    return 0;
}

C

// C program to find GCD of two numbers

#include <stdio.h>

// Recursive function to return gcd of a and b
int gcd(int a, int b)
{
    // Everything divides 0
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    // Base case
    if (a == b)
        return a;

    // a is greater
    if (a > b)
        return gcd(a - b, b);
    return gcd(a, b - a);
}

// Driver code
int main()
{
    int a = 98, b = 56;
    printf("GCD of %d and %d is %d ", a, b, gcd(a, b));
    return 0;
}

Java

// Java program to find GCD of two numbers

import java.io.*;

class Test {

    // Recursive function to return gcd of a and b
    static int gcd(int a, int b)
    {
        // Everything divides 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // Base case
        if (a == b)
            return a;

        // a is greater
        if (a > b)
            return gcd(a - b, b);
        return gcd(a, b - a);
    }

    // Driver code
    public static void main(String[] args)
    {
        int a = 98, b = 56;
        System.out.println("GCD of " + a + " and " + b
                           + " is " + gcd(a, b));
    }
}

Python3

# Python program to find GCD of two numbers


# Recursive function to return gcd of a and b
def gcd(a, b):

    # Everything divides 0
    if (a == 0):
        return b
    if (b == 0):
        return a

    # Base case
    if (a == b):
        return a

    # a is greater
    if (a > b):
        return gcd(a-b, b)
    return gcd(a, b-a)


# Driver code
if __name__ == '__main__':
    a = 98
    b = 56
    if(gcd(a, b)):
        print('GCD of', a, 'and', b, 'is', gcd(a, b))
    else:
        print('not found')

# This code is contributed by Danish Raza

C#

// C# program to find GCD of two numbers

using System;

class GFG {

    // Recursive function to return gcd of a and b
    static int gcd(int a, int b)
    {
        // Everything divides 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // Base case
        if (a == b)
            return a;

        // a is greater
        if (a > b)
            return gcd(a - b, b);

        return gcd(a, b - a);
    }

    // Driver code
    public static void Main()
    {
        int a = 98, b = 56;
        Console.WriteLine("GCD of " + a + " and " + b
                          + " is " + gcd(a, b));
    }
}

// This code is contributed by anuj_67.

Javascript

// Javascript program to find GCD of two numbers

// Recursive function to return gcd of a and b
function gcd(a, b)
{
    // Everything divides 0
    if (a == 0)
    return b;
    if (b == 0)
    return a;

    // Base case
    if (a == b)
        return a;

    // a is greater
    if (a > b)
        return gcd(a-b, b);
    return gcd(a, b-a);
}

// Driver program to test above function

    let a = 98, b = 56;
    console.log("GCD of "+ a + " and " + b + " is " + gcd(a, b));

// This code is contributed by Mayank Tyagi

PHP

<?php
// PHP program to find GCD
// of two numbers

// Recursive function to return gcd of a and b
function gcd($a, $b)
{
    // Everything divides 0
    if ($a == 0)
       return $b;
    if ($b == 0)
       return $a;

    // Base case
    if($a == $b)
        return $a ;

    // a is greater
    if($a > $b)
        return gcd( $a-$b , $b ) ;

    return gcd( $a , $b-$a ) ;
}

// Driver code
$a = 98 ;
$b = 56 ;

echo "GCD of $a and $b is ", gcd($a , $b) ;

// This code is contributed by Anivesh Tiwari
?>

Output

GCD of 98 and 56 is 14

Time Complexity: O(min(a,b))
Auxiliary Space: O(1) No space is used as it is a tail recursion.

Optimization by checking divisibility:

The above method can be optimized based on the following idea:

If we notice the previous approach, we can see at some point, one number becomes a factor of the other so instead of repeatedly subtracting till both become equal, we can check if it is a factor of the other.

Illustration:

See the below illustration for a better understanding:

Consider a = 98 and b = 56

a = 98, b = 56:

• a > b so put a = a-b and b remains same. So  a = 98-56 = 42  & b= 56. 

a = 42, b = 56:

• Since b > a, we check if b%a=0. Since answer is no, we proceed further. 
• Now b>a. So b = b-a and a remains same. So b = 56-42 = 14 & a= 42. 

a = 42, b = 14:

• Since a>b, we check if a%b=0. Now the answer is yes. 
• So we print smaller among a and b as H.C.F . i.e. 42 is  3 times of 14.

So HCF is 14. 

Below is the implementation of the above approach:

C++

// C++ program to find GCD of two numbers

#include <bits/stdc++.h>
using namespace std;

// Recursive function to return gcd of a and b
int gcd(int a, int b)
{
    // Everything divides 0
    if (a == 0)
        return b;
    if (b == 0)
        return a;

    // Base case
    if (a == b)
        return a;

    // a is greater
    if (a > b) {
        if (a % b == 0)
            return b;
        return gcd(a - b, b);
    }
    if (b % a == 0)
        return a;
    return gcd(a, b - a);
}

// Driver code
int main()
{
    int a = 98, b = 56;
    cout << "GCD of " << a << " and " << b << " is "
         << gcd(a, b);
    return 0;
}

Java

public class GCD {
    // Recursive function to return gcd of a and b
    static int gcd(int a, int b) {
        // Everything divides 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // Base case
        if (a == b)
            return a;

        // a is greater
        if (a > b) {
            if (a % b == 0)
                return b;
            return gcd(a - b, b);
        }
        if (b % a == 0)
            return a;
        return gcd(a, b - a);
    }

    // Driver code
    public static void main(String[] args) {
        int a = 98, b = 56;
        System.out.println("GCD of " + a + " and " + b + " is " + gcd(a, b));
    }
}
// This code is contributed by rambabuguphka

Python3

def gcd(a, b):
    # Everything divides 0
    if a == 0:
        return b
    if b == 0:
        return a

    # Base case
    if a == b:
        return a

    # a is greater
    if a > b:
        if a % b == 0:
            return b
        return gcd(a - b, b)
    if b % a == 0:
        return a
    return gcd(a, b - a)

# Driver code
a = 98
b = 56
print(f"GCD of {a} and {b} is {gcd(a, b)}")

C#

using System;

public class GFG
{
    // Recursive function to return gcd of a and b
    static int GCD(int a, int b)
    {
        // Everything divides 0
        if (a == 0)
            return b;
        if (b == 0)
            return a;

        // Base case
        if (a == b)
            return a;

        // a is greater
        if (a > b)
        {
            if (a % b == 0)
                return b;
            return GCD(a - b, b);
        }
        if (b % a == 0)
            return a;
        return GCD(a, b - a);
    }

    // Main method
    public static void Main(string[] args)
    {
        int a = 98, b = 56;
        Console.WriteLine("GCD of " + a + " and " + b + " is " + GCD(a, b));
    }
}

// This code is add by Avinash Wani

Javascript

// Recursive function to return gcd of a and b
function gcd(a, b) {
    // Everything divides 0
    if (a === 0) {
        return b;
    }
    if (b === 0) {
        return a;
    }

    // Base case
    if (a === b) {
        return a;
    }

    // a is greater
    if (a > b) {
        if (a % b === 0) {
            return b;
        }
        return gcd(a - b, b);
    }
    if (b % a === 0) {
        return a;
    }
    return gcd(a, b - a);
}

// Driver code
let a = 98;
let b = 56;
console.log(`GCD of ${a} and ${b} is ${gcd(a, b)}`);

Output

GCD of 98 and 56 is 14

Time Complexity: O(min(a, b))
Auxiliary Space: O(1)

Optimization using division:

Instead of the Euclidean algorithm by subtraction, a better approach can be used. We don’t perform subtraction here. we continuously divide the bigger number by the smaller number. More can be learned about this efficient solution by using the modulo operator in Euclidean algorithm.

Below is the implementation of the above approach.

C++

// C++ program to find GCD of two numbers
#include <iostream>
using namespace std;
// Recursive function to return gcd of a and b in single line
int gcd(int a, int b)
{
    return b == 0 ? a : gcd(b, a % b);
}

// Driver program to test above function
int main()
{
    int a = 98, b = 56;
    cout<<"GCD of "<<a<<" and "<<b<<" is "<<gcd(a, b);
    return 0;
}

C

// C program to find GCD of two numbers
#include <stdio.h>

// Recursive function to return gcd of a and b
int gcd(int a, int b)
{
    if (b == 0)
        return a;
    return gcd(b, a % b);
}

// Driver program to test above function
int main()
{
    int a = 98, b = 56;
    printf("GCD of %d and %d is %d ", a, b, gcd(a, b));
    return 0;
}

Java

// Java program to find GCD of two numbers
import java.io.*;

class Test
{
    // Recursive function to return gcd of a and b
    static int gcd(int a, int b)
    {
      if (b == 0)
        return a;
      return gcd(b, a % b);
    }

    // Driver method
    public static void main(String[] args)
    {
        int a = 98, b = 56;
        System.out.println("GCD of " + a +" and " + b + " is " + gcd(a, b));
    }
}

Python3

# Recursive function to return gcd of a and b
def gcd(a,b):

    # Everything divides 0
    if (b == 0):
         return a
    return gcd(b, a%b)

# Driver program to test above function
a = 98
b = 56
if(gcd(a, b)):
    print('GCD of', a, 'and', b, 'is', gcd(a, b))
else:
    print('not found')

# This code is contributed by Danish Raza

C#

// C# program to find GCD of two
// numbers
using System;

class GFG {

    // Recursive function to return
    // gcd of a and b
    static int gcd(int a, int b)
    {
       if (b == 0)
          return a;
       return gcd(b, a % b);
    }

    // Driver method
    public static void Main()
    {
        int a = 98, b = 56;
        Console.WriteLine("GCD of "
          + a +" and " + b + " is "
                      + gcd(a, b));
    }
}

// This code is contributed by anuj_67.

Javascript

<script>

// Javascript program to find GCD of two number

// Recursive function to return gcd of a and b

function gcd(a, b){

  // Everything divides 0
  if(b == 0){
    return a;
  }

  return gcd(b, a % b);
}

// Driver code
let a = 98;
let b = 56;

document.write(`GCD of ${a} and ${b} is ${gcd(a, b)}`);

// This code is contributed by _saurabh_jaiswal

</script>

PHP

<?php
// PHP program to find GCD
// of two numbers

// Recursive function to
// return gcd of a and b
function gcd($a, $b)
{
    // Everything divides 0
    if($b==0)
        return $a ;

    return gcd( $b , $a % $b ) ;
}

// Driver code
$a = 98 ;
$b = 56 ;

echo "GCD of $a and $b is ", gcd($a , $b) ;

// This code is contributed by Anivesh Tiwari
?>

Output

GCD of 98 and 56 is 14

Complexity Analysis:

Time Complexity: O(log(min(a,b))

• The derivation for this is obtained from the analysis of the worst-case scenario. 
• What we do is we ask what are the 2 least numbers that take 1 step, those would be (1,1). If we want to increase the number of steps to 2 while keeping the numbers as low as possible as we can take the numbers to be (1,2). Similarly, for 3 steps, the numbers would be (2,3), 4 would be (3,5), 5 would be (5,8). 
• So we can notice a pattern here, for the nth step the numbers would be (fib(n), fib(n+1)). So the worst-case time complexity would be O(n) where a ≥ fib(n) and b ≥ fib(n+1). 
• Now Fibonacci series is an exponentially growing series where the ratio of n^th/(n-1)^th term approaches (sqrt(5)+1)/2 which is also called the golden ratio. So we can see that the time complexity of the algorithm increases linearly as the terms grow exponentially hence the time complexity would be log(min(a,b)).

Auxiliary Space: O(log(min(a,b))

Iterative implementation for GCD of two numbers using Euclidean Algorithm:

Below is the iterative way to find the GCD of two numbers using Euclidean algorithm.

C++

// C++ program to find GCD of two numbers
#include <bits/stdc++.h>
using namespace std;

// Iterative function to return gcd of a and b
int gcd(int a, int b)
{
    while (a > 0 && b > 0) {
        if (a > b) {
            a = a % b;
        }
        else {
            b = b % a;
        }
    }
    if (a == 0) {
        return b;
    }
    return a;
}

// Driver code
int main()
{
    int a = 98, b = 56;
    cout << "GCD of " << a << " and " << b << " is "
         << gcd(a, b);
    return 0;
}

C

// C program to find GCD of two numbers
#include <stdio.h>

// Iterative function to return gcd of a and b
int gcd(int a, int b)
{
    while (a > 0 && b > 0) {
        if (a > b) {
            a = a % b;
        }
        else {
            b = b % a;
        }
    }
    if (a == 0) {
        return b;
    }
    return a;
}

// Driver code
int main()
{
    int a = 98, b = 56;
    printf("GCD of %d and %d is %d ", a, b, gcd(a, b));
    return 0;
}

Java

// Java program to find GCD of two numbers
import java.io.*;

class Test {

    // Iterative function to return gcd of a and b
    static int gcd(int a, int b)
    {
        while (a > 0 && b > 0) {
            if (a > b) {
                a = a % b;
            }
            else {
                b = b % a;
            }
        }
        if (a == 0) {
            return b;
        }
        return a;
    }

    // Driver code
    public static void main(String[] args)
    {
        int a = 98, b = 56;
        System.out.println("GCD of " + a + " and " + b
                           + " is " + gcd(a, b));
    }
}

Python3

# Itervative function to return gcd of a and b


def gcd(a, b):

    # Everything divides 0
    while(a > 0 and b > 0):
        if (a > b):
            a = a % b
        else:
            b = b % a

    if (a == 0):
        return b
    return a


# Driver code
if __name__ == '__main__':
    a = 98
    b = 56
    if(gcd(a, b)):
        print('GCD of', a, 'and', b, 'is', gcd(a, b))
    else:
        print('not found')

C#

// C# program to find GCD of two numbers
using System;

class GFG {

    // Iterative function to return gcd of a and b
    static int gcd(int a, int b)
    {
        while (a > 0 && b > 0) {
            if (a > b) {
                a = a % b;
            }
            else {
                b = b % a;
            }
        }
        if (a == 0) {
            return b;
        }
        return a;
    }

    // Driver code
    public static void Main()
    {
        int a = 98, b = 56;
        Console.WriteLine("GCD of " + a + " and " + b
                          + " is " + gcd(a, b));
    }
}

// This code is contributed by anuj_67.

Javascript

// Javascript program to find GCD of two number

// Recursive function to return gcd of a and b

function gcd(a, b){

  // Everything divides 0
      while (a > 0 && b > 0) {
        if (a > b) {
            a = a % b;
        }
        else {
            b = b % a;
        }
    }
    if (a == 0) {
        return b;
    }
    return a;
}

// Driver code
let a = 98;
let b = 56;

console.log(`GCD of ${a} and ${b} is ${gcd(a, b)}`);

// This code is contributed by _saurabh_jaiswal

Output

GCD of 98 and 56 is 14

Time Complexity: O(log(min(a,b))
Auxiliary Space: O(1)

GCD of two numbers using inbuilt function:

Languages like C++ have inbuilt functions to calculate GCD of two numbers.

Below is the implementation using inbuilt functions.

C++

// c++ program to find gcd using inbuilt functions
#include <algorithm>
#include <iostream>
using namespace std;

int main()
{
    int a = 98, b = 56;
    cout << "The gcd of a and b is " << __gcd(a, b) << endl;
    return 0;
}

Java

// JAVA program to find gcd using inbuilt functions
import java.math.BigInteger;
import java.util.*;

public class GFG {
    public static void main(String[] args)
    {
        int a = 98, b = 56;
        int gcd = gcd(a, b);
        System.out.println("The gcd of a and b is " + gcd);
    }

    public static int gcd(int a, int b)
    {
        BigInteger bigA = BigInteger.valueOf(Math.abs(a));
        BigInteger bigB = BigInteger.valueOf(Math.abs(b));
        BigInteger gcd = bigA.gcd(bigB);
        return gcd.intValue();
    }
}
// This code is contributed by Taranpreet Singh.

Python3

# Python program to find gcd using inbuilt function using math library
import math

#Driver code
if __name__ == '__main__':
  a = 98
  b = 56
  gcd_result = math.gcd(a, b) # inbuilt function gcd() using math library

  print("The gcd of a and b is", gcd_result)

# This code is contributed by guptapratik

C#

using System;

class Program
{
    static void Main()
    {
        int a = 98, b = 56;
        Console.WriteLine($"The gcd of a and b is {GCD(a, b)}");
    }

    static int GCD(int a, int b)
    {
        return Math.Abs(b == 0 ? a : GCD(b, a % b));
    }
}

Output

The gcd of a and b is 14

Time Complexity: O(log(min(a, b)))
Auxiliary Space: O(1)

Please refer GCD of more than two (or array) numbers to find HCF of more than two numbers.

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