Common Divisors of Two Numbers

Given two integer numbers, the task is to find count of all common divisors of given numbers?

Examples :

Input : a = 12, b = 24
Output: 6
// all common divisors are 1, 2, 3, 
// 4, 6 and 12

Input : a = 3, b = 17
Output: 1
// all common divisors are 1

Input : a = 20, b = 36
Output: 3
// all common divisors are 1, 2, 4

It is recommended to refer all divisors of a given number as a prerequisite of this article. 

Naive Solution
A simple solution is to first find all divisors of first number and store them in an array or hash. Then find common divisors of second number and store them. Finally print common elements of two stored arrays or hash. The key is that the magnitude of powers of prime factors of a divisor should be equal to the minimum power of two prime factors of a and b.

• Find the prime factors of a using prime factorization.
• Find the count of each prime factor of a and store it in a Hashmap.
• Prime factorize b using distinct prime factors of a.
• Then the total number of divisors would be equal to the product of (count + 1) 
  of each factor.
• count is the minimum of counts of each prime factors of a and b.
• This gives the count of all divisors of a and b.

C++

// C++ implementation of program
#include <bits/stdc++.h>
using namespace std;

// Map to store the count of each
// prime factor of a
map<int, int> ma;

// Function that calculate the count of
// each prime factor of a number
void primeFactorize(int a)
{
    for(int i = 2; i * i <= a; i += 2)
    {
        int cnt = 0;
        while (a % i == 0)
        {
            cnt++;
            a /= i;
        }
        ma[i] = cnt;
    }
    if (a > 1)
    {
        ma[a] = 1;
    }
}

// Function to calculate all common
// divisors of two given numbers
// a, b --> input integer numbers
int commDiv(int a, int b)
{

    // Find count of each prime factor of a
    primeFactorize(a);

    // stores number of common divisors
    int res = 1;

    // Find the count of prime factors
    // of b using distinct prime factors of a
    for(auto m = ma.begin();
             m != ma.end(); m++)
    {
        int cnt = 0;
        int key = m->first;
        int value = m->second;

        while (b % key == 0)
        {
            b /= key;
            cnt++;
        }

        // Prime factor of common divisor
        // has minimum cnt of both a and b
        res *= (min(cnt, value) + 1);
    }
    return res;
}

// Driver code
int main()
{
    int a = 12, b = 24;

    cout << commDiv(a, b) << endl;

    return 0;
}

// This code is contributed by divyeshrabadiya07

Java

// Java implementation of program
import java.util.*;
import java.io.*;

class GFG {
    // map to store the count of each prime factor of a
    static HashMap<Integer, Integer> ma = new HashMap<>();

    // method that calculate the count of
    // each prime factor of a number
    static void primeFactorize(int a)
    {
        for (int i = 2; i * i <= a; i += 2) {
            int cnt = 0;
            while (a % i == 0) {
                cnt++;
                a /= i;
            }
            ma.put(i, cnt);
        }
        if (a > 1)
            ma.put(a, 1);
    }

    // method to calculate all common divisors
    // of two given numbers
    // a, b --> input integer numbers
    static int commDiv(int a, int b)
    {
        // Find count of each prime factor of a
        primeFactorize(a);

        // stores number of common divisors
        int res = 1;

        // Find the count of prime factors of b using
        // distinct prime factors of a
        for (Map.Entry<Integer, Integer> m : ma.entrySet()) {
            int cnt = 0;

            int key = m.getKey();
            int value = m.getValue();

            while (b % key == 0) {
                b /= key;
                cnt++;
            }

            // prime factor of common divisor
            // has minimum cnt of both a and b
            res *= (Math.min(cnt, value) + 1);
        }
        return res;
    }

    // Driver method
    public static void main(String args[])
    {
        int a = 12, b = 24;
        System.out.println(commDiv(a, b));
    }
}

Python3

# Python3 implementation of program
import math

# Map to store the count of each
# prime factor of a
ma = {}

# Function that calculate the count of
# each prime factor of a number
def primeFactorize(a):

    sqt = int(math.sqrt(a))
    for i in range(2, sqt, 2):
        cnt = 0

        while (a % i == 0):
            cnt += 1
            a /= i

        ma[i] = cnt

    if (a > 1):
        ma[a] = 1

# Function to calculate all common
# divisors of two given numbers
# a, b --> input integer numbers
def commDiv(a, b):

    # Find count of each prime factor of a
    primeFactorize(a)

    # stores number of common divisors
    res = 1

    # Find the count of prime factors
    # of b using distinct prime factors of a
    for key, value in ma.items():
        cnt = 0

        while (b % key == 0):
            b /= key
            cnt += 1

        # Prime factor of common divisor
        # has minimum cnt of both a and b
        res *= (min(cnt, value) + 1)

    return res

# Driver code
a = 12
b = 24

print(commDiv(a, b))

# This code is contributed by Stream_Cipher

C#

// C# implementation of program
using System;
using System.Collections.Generic;

class GFG{

// Map to store the count of each
// prime factor of a
static Dictionary<int,
                  int> ma = new Dictionary<int,
                                           int>();

// Function that calculate the count of
// each prime factor of a number
static void primeFactorize(int a)
{
    for(int i = 2; i * i <= a; i += 2)
    {
        int cnt = 0;
        while (a % i == 0)
        {
            cnt++;
            a /= i;
        }
        ma.Add(i, cnt);
    }

    if (a > 1)
        ma.Add(a, 1);
}

// Function to calculate all common
// divisors of two given numbers
// a, b --> input integer numbers
static int commDiv(int a, int b)
{

    // Find count of each prime factor of a
    primeFactorize(a);

    // Stores number of common divisors
    int res = 1;

    // Find the count of prime factors
    // of b using distinct prime factors of a
    foreach(KeyValuePair<int, int> m in ma)
    {
        int cnt = 0;
        int key = m.Key;
        int value = m.Value;

        while (b % key == 0)
        {
            b /= key;
            cnt++;
        }

        // Prime factor of common divisor
        // has minimum cnt of both a and b
        res *= (Math.Min(cnt, value) + 1);
    }
    return res;
}

// Driver code
static void Main()
{
    int a = 12, b = 24;

    Console.WriteLine(commDiv(a, b));
}
}

// This code is contributed by divyesh072019

Javascript

<script>

    // JavaScript implementation of program

    // Map to store the count of each
    // prime factor of a
      let ma = new Map();

    // Function that calculate the count of
    // each prime factor of a number
    function primeFactorize(a)
    {
        for(let i = 2; i * i <= a; i += 2)
        {
            let cnt = 0;
            while (a % i == 0)
            {
                cnt++;
                a = parseInt(a / i, 10);
            }
            ma.set(i, cnt);
        }
        if (a > 1)
        {
            ma.set(a, 1);
        }
    }

    // Function to calculate all common
    // divisors of two given numbers
    // a, b --> input integer numbers
    function commDiv(a,b)
    {

        // Find count of each prime factor of a
        primeFactorize(a);

        // stores number of common divisors
        let res = 1;

        // Find the count of prime factors
        // of b using distinct prime factors of a
        ma.forEach((values,keys)=>{
            let cnt = 0;
            let key = keys;
            let value = values;

            while (b % key == 0)
            {
                b = parseInt(b / key, 10);
                cnt++;
            }

            // Prime factor of common divisor
            // has minimum cnt of both a and b
            res *= (Math.min(cnt, value) + 1);
        })
        return res;
    }

    // Driver code

    let a = 12, b = 24;

    document.write(commDiv(a, b));

</script>

Output:

6

Time Complexity: O(?n log n) 
Auxiliary Space: O(n)

Efficient Solution – 
A better solution is to calculate the greatest common divisor (gcd) of given two numbers, and then count divisors of that gcd. 

C++

// C++ implementation of program
#include <bits/stdc++.h>
using namespace std;

// Function to calculate gcd of two numbers
int gcd(int a, int b)
{
    if (a == 0)
        return b;
    return gcd(b % a, a);
}

// Function to calculate all common divisors
// of two given numbers
// a, b --> input integer numbers
int commDiv(int a, int b)
{
    // find gcd of a, b
    int n = gcd(a, b);

    // Count divisors of n.
    int result = 0;
    for (int i = 1; i <= sqrt(n); i++) {
        // if 'i' is factor of n
        if (n % i == 0) {
            // check if divisors are equal
            if (n / i == i)
                result += 1;
            else
                result += 2;
        }
    }
    return result;
}

// Driver program to run the case
int main()
{
    int a = 12, b = 24;
    cout << commDiv(a, b);
    return 0;
}

Java

// Java implementation of program

class Test {
    // method to calculate gcd of two numbers
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;

        return gcd(b % a, a);
    }
    // method to calculate all common divisors
    // of two given numbers
    // a, b --> input integer numbers
    static int commDiv(int a, int b)
    {
        // find gcd of a, b
        int n = gcd(a, b);

        // Count divisors of n.
        int result = 0;
        for (int i = 1; i <= Math.sqrt(n); i++) {
            // if 'i' is factor of n
            if (n % i == 0) {
                // check if divisors are equal
                if (n / i == i)
                    result += 1;
                else
                    result += 2;
            }
        }
        return result;
    }

    // Driver method
    public static void main(String args[])
    {
        int a = 12, b = 24;
        System.out.println(commDiv(a, b));
    }
}

Python3

# Python implementation of program
from math import sqrt


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

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

# Function to calculate all common divisors
# of two given numbers
# a, b --> input integer numbers
def commDiv(a, b):

    # find GCD of a, b
    n = gcd(a, b)

    # Count divisors of n
    result = 0
    for i in range(1,int(sqrt(n))+1):

        # if i is a factor of n
        if n % i == 0:

            # check if divisors are equal
            if n/i == i:
                result += 1
            else:
                result += 2

    return result

# Driver program to run the case
if __name__ == "__main__":
    a = 12
    b = 24;
    print(commDiv(a, b))

C#

// C# implementation of program
using System;

class GFG {

    // method to calculate gcd
    // of two numbers
    static int gcd(int a, int b)
    {
        if (a == 0)
            return b;

        return gcd(b % a, a);
    }

    // method to calculate all
    // common divisors of two
    // given numbers a, b -->
    // input integer numbers
    static int commDiv(int a, int b)
    {

        // find gcd of a, b
        int n = gcd(a, b);

        // Count divisors of n.
        int result = 0;
        for (int i = 1; i <= Math.Sqrt(n); i++) {

            // if 'i' is factor of n
            if (n % i == 0) {

                // check if divisors are equal
                if (n / i == i)
                    result += 1;
                else
                    result += 2;
            }
        }

        return result;
    }

    // Driver method
    public static void Main(String[] args)
    {

        int a = 12, b = 24;

        Console.Write(commDiv(a, b));
    }
}

// This code contributed by parashar.

PHP

<?php
// PHP implementation of program

// Function to calculate
// gcd of two numbers
function gcd($a, $b)
{
    if ($a == 0)
        return $b;
    return gcd($b % $a, $a);
}

// Function to calculate all common
// divisors of two given numbers
// a, b --> input integer numbers
function commDiv($a, $b)
{
    // find gcd of a, b
    $n = gcd($a, $b);

    // Count divisors of n.
    $result = 0;
    for ($i = 1; $i <= sqrt($n);
                 $i++)
    {
        // if 'i' is factor of n
        if ($n % $i == 0)
        {
            // check if divisors
            // are equal
            if ($n / $i == $i)
                $result += 1;
            else
                $result += 2;
        }
    }
    return $result;
}

// Driver Code
$a = 12; $b = 24;
echo(commDiv($a, $b));

// This code is contributed by Ajit.
?>

Javascript

<script>
    // Javascript implementation of program

    // Function to calculate gcd of two numbers
    function gcd(a, b)
    {
        if (a == 0)
            return b;
        return gcd(b % a, a);
    }

    // Function to calculate all common divisors
    // of two given numbers
    // a, b --> input integer numbers
    function commDiv(a, b)
    {
        // find gcd of a, b
        let n = gcd(a, b);

        // Count divisors of n.
        let result = 0;
        for (let i = 1; i <= Math.sqrt(n); i++) {
            // if 'i' is factor of n
            if (n % i == 0) {
                // check if divisors are equal
                if (n / i == i)
                    result += 1;
                else
                    result += 2;
            }
        }
        return result;
    }

    let a = 12, b = 24;
    document.write(commDiv(a, b));

</script>

Output :

6

Time complexity: O(n^1/2) where n is the gcd of two numbers.
Auxiliary Space: O(1)

Another Approach:

1. Define a function “gcd” that takes two integers “a” and “b” and returns their greatest common divisor (GCD) using the Euclidean algorithm.
2. Define a function “count_common_divisors” that takes two integers “a” and “b” and counts the number of common divisors of “a” and “b” using their GCD.
3. Calculate the GCD of “a” and “b” using the “gcd” function.
4. Initialize a counter “count” to 0.
5. Loop through all possible divisors of the GCD of “a” and “b” from 1 to the square root of the GCD.
6. If the current divisor divides the GCD evenly, increment the counter by 2 (because both “a” and “b” are divisible by the divisor).
7. If the square of the current divisor equals the GCD, decrement the counter by 1 (because we’ve already counted this divisor once).
8. Return the final count of common divisors.
9. In the main function, define two integers “a” and “b” and call the “count_common_divisors” function with these integers.
10. Print the number of common divisors of “a” and “b” using the printf function.

C

#include <stdio.h>

int gcd(int a, int b) {
    if(b == 0) {
        return a;
    }
    return gcd(b, a % b);
}

int count_common_divisors(int a, int b) {
    int gcd_ab = gcd(a, b);
    int count = 0;

    for(int i = 1; i * i <= gcd_ab; i++) {
        if(gcd_ab % i == 0) {
            count += 2;
            if(i * i == gcd_ab) {
                count--;
            }
        }
    }

    return count;
}

int main() {
    int a = 12;
    int b = 18;

    int common_divisors = count_common_divisors(a, b);

    printf("The number of common divisors of %d and %d is %d.\n", a, b, common_divisors);

    return 0;
}

C++

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

int gcd(int a, int b) {
    if(b == 0) {
        return a;
    }
    return gcd(b, a % b);
}

int count_common_divisors(int a, int b) {
    int gcd_ab = gcd(a, b);
    int count = 0;

    for(int i = 1; i * i <= gcd_ab; i++) {
        if(gcd_ab % i == 0) {
            count += 2;
            if(i * i == gcd_ab) {
                count--;
            }
        }
    }

    return count;
}

int main() {
    int a = 12;
    int b = 18;

    int common_divisors = count_common_divisors(a, b);
    cout<<"The number of common divisors of "<<a<<" and "<<b<<" is "<<common_divisors<<"."<<endl;
    return 0;
}

Java

import java.util.*;

public class Main {

  public static int gcd(int a, int b) {
    if(b == 0) {
      return a;
    }
    return gcd(b, a % b);
  }

  public static int countCommonDivisors(int a, int b) {
    int gcd_ab = gcd(a, b);
    int count = 0;

    for(int i = 1; i * i <= gcd_ab; i++) {
      if(gcd_ab % i == 0) {
        count += 2;
        if(i * i == gcd_ab) {
          count--;
        }
      }
    }

    return count;
  }

  public static void main(String[] args) {
    int a = 12;
    int b = 18;

    int commonDivisors = countCommonDivisors(a, b);
    System.out.println("The number of common divisors of " + a + " and " + b + " is " + commonDivisors + ".");
  }
}

Python3

import math

def gcd(a, b):
    if b == 0:
        return a
    return gcd(b, a % b)

def count_common_divisors(a, b):
    gcd_ab = gcd(a, b)
    count = 0

    for i in range(1, int(math.sqrt(gcd_ab)) + 1):
        if gcd_ab % i == 0:
            count += 2
            if i * i == gcd_ab:
                count -= 1

    return count

a = 12
b = 18

common_divisors = count_common_divisors(a, b)
print("The number of common divisors of", a, "and", b, "is", common_divisors, ".")

# This code is contributed by Prajwal Kandekar

C#

using System;

public class MainClass
{
    public static int GCD(int a, int b)
    {
        if (b == 0)
        {
            return a;
        }
        return GCD(b, a % b);
    }

    public static int CountCommonDivisors(int a, int b)
    {
        int gcd_ab = GCD(a, b);
        int count = 0;

        for (int i = 1; i * i <= gcd_ab; i++)
        {
            if (gcd_ab % i == 0)
            {
                count += 2;
                if (i * i == gcd_ab)
                {
                    count--;
                }
            }
        }

        return count;
    }

    public static void Main()
    {
        int a = 12;
        int b = 18;

        int commonDivisors = CountCommonDivisors(a, b);
        Console.WriteLine("The number of common divisors of {0} and {1} is {2}.", a, b, commonDivisors);
    }
}

Javascript

// Function to calculate the greatest common divisor of
// two integers a and b using the Euclidean algorithm
function gcd(a, b) {
    if(b === 0) {
        return a;
    }
    return gcd(b, a % b);
}

// Function to count the number of common divisors of two integers a and b
function count_common_divisors(a, b) {
    let gcd_ab = gcd(a, b);
    let count = 0;

    for(let i = 1; i * i <= gcd_ab; i++) {
        if(gcd_ab % i === 0) {
            count += 2;
            if(i * i === gcd_ab) {
                count--;
            }
        }
    }

    return count;
}

let a = 12;
let b = 18;

let common_divisors = count_common_divisors(a, b);
console.log(`The number of common divisors of ${a} and ${b} is ${common_divisors}.`);

Output

The number of common divisors of 12 and 18 is 4.

The time complexity of the gcd() function is O(log(min(a, b))), as it uses Euclid’s algorithm which takes logarithmic time with respect to the smaller of the two numbers.

The time complexity of the count_common_divisors() function is O(sqrt(gcd(a, b))), as it iterates up to the square root of the gcd of the two numbers.

The space complexity of both functions is O(1), as they only use a constant amount of memory regardless of the input size.