Check if given number has 7 divisors

Given a number N, the task is to check whether N has 7 divisors or not.

Examples:

Input: 64
Output: 1 
Explanation: 1, 2, 4, 8, 16, 32, 64 -> 7 divisors so output is 1

Input: 100
Output: 0
Explanation: 1, 2, 4, 5, 10, 20, 25, 50, 100 -> 8 divisors so output is 0

Input: 729
Output: 1
Explanation: 1, 2, 4, 8, 16, 32, 64 -> 7 divisors so output is 1

Approach: The problem can be solved based on the following mathematical observation:

The prime factorization of a number is:
N = p₁^e1 * p₂^e2 * . . . * p_n^en
So number of divisors = ( e₁ + 1 ) * (e₂ + 1) *. . . * (e_n + 1).

In this case, number of divisors = 7 .
So, ( e₁ + 1 ) * (e₂ + 1) *. . . * (e_n + 1) = 7

7 is multiplication of at most 2 natural numbers. 
So, it can be only written as ( e₁ + 1 ) * (e₂ + 1) = 1 * 7    
So, e₁ = 0  &  e₂ = 6  from above equation.

So, it is clear that for 7 divisors only one case is possible and that is (prime number)⁶. Follow the steps to solve the problem:

• Check if N^(1/6) is a prime number or not.
• If it is prime number then output “Yes”.
• Otherwise, the output will be “No”.

Below is the implementation of the above approach: 

C++

// C++ code to implement the approach

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

// Function to check number of
// divisors are 7 or not
int sevenDivisors(int N)
{
    // Using power function to get 6th Root
    int k = pow(N, 1 / 6.);

    // Using power function to get
    // 6th power of k
    int res = pow(k, 6);

    // If res is equal to given number
    // N then return 1
    if (N == res)
        return 1;
    return 0;
}

// Driver code
int main()
{
    int N = 64;

    // Function call
    bool ans = sevenDivisors(N);
    if (ans)
        cout << "Yes";
    else
        cout << "No";
    return 0;
}

Java

// Java code to implement the approach
import java.io.*;

class GFG {

  public boolean sevenDivisors(int N)
  {

    // Using power function to get 6th Root
    int k = (int)Math.pow(N, 1 / 6.);

    // Using power function to get
    // 6th power of k
    int res = (int)Math.pow(k, 6);

    // If res is equal to given number
    // N then return 1
    if (N == res)
      return true;
    return false;
  }
  public static void main(String[] args)
  {
    int N = 64;

    // Function call
    GFG g1 = new GFG();
    boolean ans = g1.sevenDivisors(N);
    if (ans)
      System.out.println("Yes");
    else
      System.out.println("No");
  }
}

// This code is contributed by patildhanu4111999.

Python3

# Python3 code to implement the approach

# Function to check number of
# divisors are 7 or not
def sevenDivisors(N) :

    # Using power function to get 6th Root
    k = pow(N, 1 / 6);

    # Using power function to get
    # 6th power of k
    res = pow(k, 6);

    # If res is equal to given number
    # N then return 1
    if (N == res) :
        return 1;

    return 0;

# Driver code
if __name__ == "__main__" :

    N = 64;

    # Function call
    ans = sevenDivisors(N);
    if (ans) :
        print("Yes");
    else :
        print("No");

    # This code is contributed by AnkThon

C#

// C# code to implement the approach
using System;

public class GFG {

  public bool sevenDivisors(int N)
  {

    // Using power function to get 6th Root
    int k = (int)Math.Pow(N, 1 / 6.0);

    // Using power function to get
    // 6th power of k
    int res = (int)Math.Pow(k, 6);

    // If res is equal to given number
    // N then return 1
    if (N == res)
      return true;
    return false;
  }
  public static void Main(String[] args)
  {
    int N = 64;

    // Function call
    GFG g1 = new GFG();
    bool ans = g1.sevenDivisors(N);
    if (ans)
      Console.WriteLine("Yes");
    else
      Console.WriteLine("No");
  }
}

// This code contributed by shikhasingrajput

Javascript

<script>
// Javascript code to implement the approach

// Function to check number of
// divisors are 7 or not
function sevenDivisors(N)
{
    // Using power function to get 6th Root
    let k = Math.pow(N, 1 / 6.);

    // Using power function to get
    // 6th power of k
    let res = Math.pow(k, 6);

    // If res is equal to given number
    // N then return 1
    if (N == res)
        return 1;
    return 0;
}

// Driver code

    let N = 64;

    // Function call
    let ans = sevenDivisors(N);
    if (ans)
         document.write('YES',"</br>");
    else
         document.write('No',"</br>");
    </script>

Output

Yes

Time Complexity: O(logN)
Auxiliary Space: O(1)