Given two integer 
and 
, the task is to find the count of common factors of two numbers where factors are prime.
Examples:
Input: A = 6, B = 12
Output: 2
2 and 3 are the only common prime divisors of 6 and 12
Input: A = 4, B = 8
Output: 1
Naive Approach: Iterate from 1 to min(A, B) and check whether i is prime and a factor of both A and B, if yes then increment the counter.
Efficient Approach is to do following:
- Find Greatest Common Divisor (gcd) of the given numbers.
- Find prime factors of the GCD.
Below is the implementation of the above approach:
C++
// CPP program to count common prime factors// of a and b.#include <bits/stdc++.h>using namespace std;// A function to count all prime factors of// a given number xint countPrimeFactors(int x){ int res = 0; if (x % 2 == 0) { res++; // Print the number of 2s that divide x while (x % 2 == 0) x = x / 2; } // x must be odd at this point. So we // can skip one element (Note i = i +2) for (int i = 3; i <= sqrt(x); i = i + 2) { if (x % i == 0) { // While i divides x, print i and // divide x res++; while (x % i == 0) x = x / i; } } // This condition is to handle the case // when x is a prime number greater than 2 if (x > 2) res++; return res;}// Count of common prime factorsint countCommonPrimeFactors(int a, int b){ // Get the GCD of the given numbers int gcd = __gcd(a, b); // Count prime factors in GCD return countPrimeFactors(gcd);}// Driver codeint main(){ int a = 6, b = 12; cout << countCommonPrimeFactors(a, b); return 0;} |
C
// C program to count common prime factors// of a and b.#include <stdio.h>#include <math.h>// A function to count all prime factors of// a given number xint countPrimeFactors(int x){ int res = 0; if (x % 2 == 0) { res++; // Print the number of 2s that divide x while (x % 2 == 0) x = x / 2; } // x must be odd at this point. So we // can skip one element (Note i = i +2) for (int i = 3; i <= sqrt(x); i = i + 2) { if (x % i == 0) { // While i divides x, print i and // divide x res++; while (x % i == 0) x = x / i; } } // This condition is to handle the case // when x is a prime number greater than 2 if (x > 2) res++; return res;}// Count of common prime factorsint countCommonPrimeFactors(int a, int b){ int gcd, i; // Get the GCD of the given numbers for(i = 1; i <= a && i <= b; ++i) { // Checks if i is factor of both integers if(a % i == 0 && b % i == 0) gcd = i; } // Count prime factors in GCD return countPrimeFactors(gcd);}// Driver codeint main(){ int a = 6, b = 12; printf("%d",countCommonPrimeFactors(a, b)); return 0;}// This code is contributed by kothavvsaakash. |
Java
// Java program to count common prime factors // of a and b.import java.io.*;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); } // A function to count all prime factors of// a given number x static int countPrimeFactors(int x){ int res = 0; if (x % 2 == 0) { res++; // Print the number of 2s that divide x while (x % 2 == 0) x = x / 2; } // x must be odd at this point. So we // can skip one element (Note i = i +2) for (int i = 3; i <= Math.sqrt(x); i = i + 2) { if (x % i == 0) { // While i divides x, print i and // divide x res++; while (x % i == 0) x = x / i; } } // This condition is to handle the case // when x is a prime number greater than 2 if (x > 2) res++; return res;}// Count of common prime factorsstatic int countCommonPrimeFactors(int a, int b){ // Get the GCD of the given numbers int gcd = __gcd(a, b); // Count prime factors in GCD return countPrimeFactors(gcd);}// Driver code public static void main (String[] args) { int a = 6, b = 12; System.out.println(countCommonPrimeFactors(a, b)); }}// This code is contributed by inder_verma.. |
Python3
# Python 3 program to count common prime # factors of a and b.from math import sqrt,gcd# A function to count all prime # factors of a given number xdef countPrimeFactors(x): res = 0 if (x % 2 == 0): res += 1 # Print the number of 2s that divide x while (x % 2 == 0): x = x / 2 # x must be odd at this point. So we # can skip one element (Note i = i +2) k = int(sqrt(x)) + 1 for i in range(3, k, 2): if (x % i == 0): # While i divides x, print i # and divide x res += 1 while (x % i == 0): x = x / i # This condition is to handle the # case when x is a prime number # greater than 2 if (x > 2): res += 1 return res# Count of common prime factorsdef countCommonPrimeFactors(a, b): # Get the GCD of the given numbers gcd__ = gcd(a, b) # Count prime factors in GCD return countPrimeFactors(gcd__)# Driver codeif __name__ == '__main__': a = 6 b = 12 print(countCommonPrimeFactors(a, b)) # This code is contributed by# Surendra_Gangwar |
C#
// C# program to count common prime factors // of a and b. 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); } // A function to count all prime factors of // a given number x static int countPrimeFactors(int x) { int res = 0; if (x % 2 == 0) { res++; // Print the number of 2s that divide x while (x % 2 == 0) x = x / 2; } // x must be odd at this point. So we // can skip one element (Note i = i +2) for (int i = 3; i <= Math.Sqrt(x); i = i + 2) { if (x % i == 0) { // While i divides x, print i and // divide x res++; while (x % i == 0) x = x / i; } } // This condition is to handle the case // when x is a prime number greater than 2 if (x > 2) res++; return res; } // Count of common prime factors static int countCommonPrimeFactors(int a, int b) { // Get the GCD of the given numbers int gcd = __gcd(a, b); // Count prime factors in GCD return countPrimeFactors(gcd); } // Driver code public static void Main() { int a = 6, b = 12; Console.WriteLine(countCommonPrimeFactors(a, b)); } // This code is contributed by Ryuga } |
PHP
<?php// PHP program to count common // prime factors of a and b. // 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 - $a)); } // A function to count all prime // factors of a given number x function countPrimeFactors($x) { $res = 0; if ($x % 2 == 0) { $res++; // Print the number of 2s that // divide x while ($x % 2 == 0) $x = $x / 2; } // x must be odd at this point. So we // can skip one element (Note i = i +2) for ($i = 3; $i <= sqrt($x); $i = $i + 2) { if ($x % $i == 0) { // While i divides x, print i // and divide x $res++; while ($x % $i == 0) $x = $x / $i; } } // This condition is to handle the case // when x is a prime number greater than 2 if ($x > 2) $res++; return $res; } // Count of common prime factors function countCommonPrimeFactors($a, $b) { // Get the GCD of the given numbers $gcd = __gcd($a, $b); // Count prime factors in GCD return countPrimeFactors($gcd); } // Driver code $a = 6;$b = 12; echo (countCommonPrimeFactors($a, $b)); // This code is contributed by akt_mit..?> |
Javascript
<script> // Javascript program to count // common prime factors of a and b. // 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); } // A function to count all prime factors of // a given number x function countPrimeFactors(x) { let res = 0; if (x % 2 == 0) { res++; // Print the number of 2s that divide x while (x % 2 == 0) x = parseInt(x / 2, 10); } // x must be odd at this point. So we // can skip one element (Note i = i +2) for (let i = 3; i <= Math.sqrt(x); i = i + 2) { if (x % i == 0) { // While i divides x, print i and // divide x res++; while (x % i == 0) x = parseInt(x / i, 10); } } // This condition is to handle the case // when x is a prime number greater than 2 if (x > 2) res++; return res; } // Count of common prime factors function countCommonPrimeFactors(a, b) { // Get the GCD of the given numbers let gcd = __gcd(a, b); // Count prime factors in GCD return countPrimeFactors(gcd); } let a = 6, b = 12; document.write(countCommonPrimeFactors(a, b)); </script> |
Output:
2
Time Complexity: O(sqrtn*logn)
Auxiliary Space: O(1)
If there are multiple queries for counting common divisors, we can further optimize above code using Prime Factorization using Sieve O(log n) for multiple queries
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