Number of ways to divide a N elements equally into group of at least 2

Given an integer N denoting the number of elements, the task is to find the number of ways to divide these elements equally into groups such that each group has at least 2 elements.

Examples:

Input: N = 2 
Output: 1
Explanation: There can be only one group.

Input: N = 10
Output: 3
Explanation: There are 3 ways to divide elements:
One group having all the 10 elements.
Two groups where each group has 5 elements.
Five groups where each group has 2 elements.

Approach: The above problem can be solved using the below-given brute force approach. In every iteration of the loop, i represents the number of groups. If N is completely divisible by i, hence, the elements can be equally divided among groups. Follow the steps below to solve the problem:

• Declare variable ways and initialize it by 0.
• Iterate over the range [1, N/2] using the variable i and perform the following tasks:
  • Check if N is completely divisible by i.
  • If yes, then increment ways by 1.
• After performing the above steps, print the value of ways as the answer.

Below is the implementation of the above approach:

C++

// C++ program for the given approach
#include <iostream>
using namespace std;

// Function to find the number of ways
int numberofWays(int N)
{
    // Variable to store the number of ways
    int ways = 0;
    int i;

    // Loop to find total number of ways
    for (i = 1; i <= N / 2; i++) {
        if (N % i == 0)
            ways++;
    }

    // Returning the number of ways
    return ways;
}

// Driver Code
int main()
{

    // Declaring and initialising N
    int N = 10;

    // Function call
    int ans = numberofWays(N);

    // Displaying the answer on screen
    cout << ans;
    return 0;
}

Java

// Java program for the above approach
import java.io.*;
import java.lang.*;
import java.util.*;

class GFG {

  // Function to find the number of ways
  static int numberofWays(int N)
  {
    // Variable to store the number of ways
    int ways = 0;
    int i;

    // Loop to find total number of ways
    for (i = 1; i <= N / 2; i++) {
      if (N % i == 0)
        ways++;
    }

    // Returning the number of ways
    return ways;
  }

  public static void main (String[] args)
  {

    // Declaring and initialising N
    int N = 10;

    // Function call
    int ans = numberofWays(N);

    // Displaying the answer on screen
    System.out.print(ans);
  }
}

// This code is contributed by hrithikgarg03188

Python3

# Python code for the above approach

# Function to find the number of ways
def numberofWays(N):

    # Variable to store the number of ways
    ways = 0;
    i = None

    # Loop to find total number of ways
    for i in range(1, (N // 2) + 1):
        if (N % i == 0):
            ways += 1

    # Returning the number of ways
    return ways;

# Driver Code

# Declaring and initialising N
N = 10;

# Function call
ans = numberofWays(N);

# Displaying the answer on screen
print(ans);

# This code is contributed by Saurabh Jaiswal

C#

// C# program for the above approach
using System;

class GFG {

  // Function to find the number of ways
  static int numberofWays(int N)
  {
    // Variable to store the number of ways
    int ways = 0;
    int i;

    // Loop to find total number of ways
    for (i = 1; i <= N / 2; i++) {
      if (N % i == 0)
        ways++;
    }

    // Returning the number of ways
    return ways;
  }

  public static void Main(string[] args)
  {

    // Declaring and initialising N
    int N = 10;

    // Function call
    int ans = numberofWays(N);

    // Displaying the answer on screen
    Console.WriteLine(ans);
  }
}

// This code is contributed by ukasp.

Javascript

<script>
        // JavaScript code for the above approach

        // Function to find the number of ways
        function numberofWays(N)
        {

            // Variable to store the number of ways
            let ways = 0;
            let i;

            // Loop to find total number of ways
            for (i = 1; i <= Math.floor(N / 2); i++) {
                if (N % i == 0)
                    ways++;
            }

            // Returning the number of ways
            return ways;
        }

        // Driver Code

        // Declaring and initialising N
        let N = 10;

        // Function call
        let ans = numberofWays(N);

        // Displaying the answer on screen
        document.write(ans);

  // This code is contributed by Potta Lokesh
    </script>

Output

3

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