Skip to content
Related Articles

Related Articles

Improve Article

Maximum possible prime divisors that can exist in numbers having exactly N divisors

  • Last Updated : 30 May, 2021

Given an integer N which denotes the number of divisors of any number, the task is to find the maximum prime divisors that are possible in number having N divisors.

Examples: 

Input: N = 4 
Output: 2

Input: N = 8 
Output:
 

Approach: The idea is to find the prime factorization of the number N, then the sum of the powers of the prime divisors is the maximum possible prime divisors of a number can have with N divisors.



For Example:  

Let the number of divisors of number be 4,

Then the possible numbers can be 6, 10, 15,...
Divisors of 6 = 1, 2, 3, 6

Total number of prime-divisors = 2 (2, 3)

Prime Factorization of 4 = 22
Sum of powers of prime factors = 2

Below is the implementation of the above approach: 

C++




// C++ implementation to find the
// maximum possible prime divisor
// of a number can have N divisors
 
#include <iostream>
 
using namespace std;
 
#define ll long long int
 
// Function to find the
// maximum possible prime divisors
// of a number can have with N divisors
void findMaxPrimeDivisor(int n){
     
    int max_possible_prime = 0;
 
    // Number of time number
    // divided by 2
    while (n % 2 == 0) {
        max_possible_prime++;
        n = n / 2;
    }
 
    // Divide by other prime numbers
    for (int i = 3; i * i <= n; i = i + 2) {
        while (n % i == 0) {
            max_possible_prime++;
            n = n / i;
        }
    }
 
    // If the last number of also
    // prime then also include it
    if (n > 2) {
        max_possible_prime++;
    }
 
    cout << max_possible_prime << "\n";
}
 
// Driver Code
int main()
{
 
    int n = 4;
     
    // Function Call
    findMaxPrimeDivisor(n);
    return 0;
}


Java




// Java implementation to find the
// maximum possible prime divisor
// of a number can have N divisors
import java.util.*;
 
class GFG{
 
// Function to find the
// maximum possible prime divisors
// of a number can have with N divisors
static void findMaxPrimeDivisor(int n)
{
    int max_possible_prime = 0;
 
    // Number of time number
    // divided by 2
    while (n % 2 == 0)
    {
        max_possible_prime++;
        n = n / 2;
    }
 
    // Divide by other prime numbers
    for(int i = 3; i * i <= n; i = i + 2)
    {
       while (n % i == 0)
       {
           max_possible_prime++;
           n = n / i;
       }
    }
 
    // If the last number of also
    // prime then also include it
    if (n > 2)
    {
        max_possible_prime++;
    }
    System.out.print(max_possible_prime + "\n");
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 4;
     
    // Function Call
    findMaxPrimeDivisor(n);
}
}
 
// This code is contributed by amal kumar choubey


Python3




# Python3 implementation to find the
# maximum possible prime divisor
# of a number can have N divisors
 
# Function to find the maximum
# possible prime divisors of a
# number can have with N divisors
def findMaxPrimeDivisor(n):
     
    max_possible_prime = 0
     
    # Number of time number
    # divided by 2
    while (n % 2 == 0):
        max_possible_prime += 1
        n = n // 2
         
    # Divide by other prime numbers
    i = 3
    while(i * i <= n):
        while (n % i == 0):
             
            max_possible_prime += 1
            n = n // i
        i = i + 2
         
    # If the last number of also
    # prime then also include it
    if (n > 2):
        max_possible_prime += 1
     
    print(max_possible_prime)
 
# Driver Code
n = 4
 
# Function Call
findMaxPrimeDivisor(n)
 
# This code is contributed by SHUBHAMSINGH10


C#




// C# implementation to find the
// maximum possible prime divisor
// of a number can have N divisors
using System;
 
class GFG{
 
// Function to find the
// maximum possible prime divisors
// of a number can have with N divisors
static void findMaxPrimeDivisor(int n)
{
    int max_possible_prime = 0;
 
    // Number of time number
    // divided by 2
    while (n % 2 == 0)
    {
        max_possible_prime++;
        n = n / 2;
    }
 
    // Divide by other prime numbers
    for(int i = 3; i * i <= n; i = i + 2)
    {
       while (n % i == 0)
       {
           max_possible_prime++;
           n = n / i;
       }
    }
 
    // If the last number of also
    // prime then also include it
    if (n > 2)
    {
        max_possible_prime++;
    }
    Console.Write(max_possible_prime + "\n");
}
 
// Driver Code
public static void Main(String[] args)
{
    int n = 4;
     
    // Function Call
    findMaxPrimeDivisor(n);
}
}
 
// This code is contributed by amal kumar choubey


Javascript




<script>
 
// JavaScript implementation to find the
// maximum possible prime divisor
// of a number can have N divisors
 
// Function to find the maximum
// possible prime divisors of a
// number can have with N divisors
function findMaxPrimeDivisor(n)
{
    let max_possible_prime = 0;
 
    // Number of time number
    // divided by 2
    while (n % 2 == 0)
    {
        max_possible_prime++;
        n = Math.floor(n / 2);
    }
 
    // Divide by other prime numbers
    for(let i = 3; i * i <= n; i = i + 2)
    {
        while (n % i == 0)
        {
            max_possible_prime++;
            n = Math.floor(n / i);
        }
    }
 
    // If the last number of also
    // prime then also include it
    if (n > 2)
    {
        max_possible_prime++;
    }
    document.write(max_possible_prime + "\n");
}
 
// Driver Code
let n = 4;
 
// Function Call
findMaxPrimeDivisor(n);
 
// This code is contributed by Surbhi Tyagi.
 
</script>


Output: 

2

 

Time Complexity: O(sqrt(N) * logN )

Auxiliary Space: O(1)

Attention reader! Don’t stop learning now. Get hold of all the important DSA concepts with the DSA Self Paced Course at a student-friendly price and become industry ready.  To complete your preparation from learning a language to DS Algo and many more,  please refer Complete Interview Preparation Course.

In case you wish to attend live classes with experts, please refer DSA Live Classes for Working Professionals and Competitive Programming Live for Students.




My Personal Notes arrow_drop_up
Recommended Articles
Page :