Maximum size of subset such that product of all subset elements is a factor of N

Given an integer N and an array arr[] having M integers, the task is to find the maximum size of the subset such that the product of all elements of the subset is a factor of N.

Examples:

Input: N = 12, arr[] = {2, 3, 4}
Output: 2
Explanation: The given array 5 subsets such that the product of all elements of the subset is a factor of 12. They are {2}, {3}, {4}, {2, 3} as (2 * 3) = 6, and {3, 4} as (3 * 4) = 12. Therefore, the maximum size of the valid subset is 2.

Input: N = 64, arr[] = {1, 2, 4, 8, 16, 32}
Output: 4

Approach: The given problem can be solved using recursion by traversing over all the subsets of the given array arr[] and keeping track of the size of the subsets such that N % (product of subset elements) = 0. Also for the product of the subset elements to be a factor of N, all individual elements of the array arr[] must also be a factor of N. Therefore, the above problem can be solved using the following steps:

• Create a recursive function maximizeSubset(), which calculates the maximum size of the required subset.
• If all the elements of the given array arr[] have been traversed, return 0 which is the base case.
• Iterate over all elements of the array arr[] and if N % arr[i] = 0, include arr[i] in a subset and recursively call for N = N/arr[i] for the remaining array elements.

Below is the implementation of the above approach:

C++

// C++ program for the above approach

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

// Function to find the maximum size of
// the subset such that the product of
// subset elements is a factor of N
int maximizeSubset(int N, int arr[],
                   int M, int x = 0)
{
    // Base Case
    if (x == M) {
        return 0;
    }

    // Stores maximum size of valid subset
    int ans = 0;

    // Traverse the given array
    for (int i = x; i < M; i++) {

        // If N % arr[i] = 0, include arr[i]
        // in a subset and recursively call
        // for the remaining array integers
        if (N % arr[i] == 0) {
            ans = max(
                ans, maximizeSubset(
                         N / arr[i], arr,
                         M, x + 1)
                         + 1);
        }
    }

    // Return Answer
    return ans;
}

// Driver Code
int main()
{
    int N = 64;
    int arr[] = { 1, 2, 4, 8, 16, 32 };
    int M = sizeof(arr) / sizeof(arr[0]);

    cout << maximizeSubset(N, arr, M);

    return 0;
}

Java

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

class GFG {

    // Function to find the maximum size of
    // the subset such that the product of
    // subset elements is a factor of N
    static int maximizeSubset(int N, int[] arr, int M,
                              int x)
    {
        // Base Case
        if (x == M) {
            return 0;
        }

        // Stores maximum size of valid subset
        int ans = 0;

        // Traverse the given array
        for (int i = x; i < M; i++) {

            // If N % arr[i] = 0, include arr[i]
            // in a subset and recursively call
            // for the remaining array integers
            if (N % arr[i] == 0) {
                ans = Math.max(ans,
                               maximizeSubset(N / arr[i],
                                              arr, M, x + 1)
                                   + 1);
            }
        }

        // Return Answer
        return ans;
    }

    // Driver Code
    public static void main(String[] args)
    {
        int N = 64;
        int[] arr = { 1, 2, 4, 8, 16, 32 };
        int M = arr.length;

        System.out.println(maximizeSubset(N, arr, M, 0));
    }
}

// This code is contributed by ukasp.

Python3

# Python Program to implement
# the above approach

# Function to find the maximum size of
# the subset such that the product of
# subset elements is a factor of N
def maximizeSubset(N, arr, M, x=0):

    # Base Case
    if (x == M):
        return 0

    # Stores maximum size of valid subset
    ans = 0

    # Traverse the given array
    for i in range(x, M):

        # If N % arr[i] = 0, include arr[i]
        # in a subset and recursively call
        # for the remaining array integers
        if (N % arr[i] == 0):
            ans = max(
                ans, maximizeSubset(
                    N // arr[i], arr,
                    M, x + 1)
                + 1)

    # Return Answer
    return ans


# Driver Code
N = 64
arr = [1, 2, 4, 8, 16, 32]
M = len(arr)

print(maximizeSubset(N, arr, M))

# This code is contributed by Saurabh jaiswal

C#

// C# program for the above approach
using System;
using System.Collections.Generic;

class GFG{

// Function to find the maximum size of
// the subset such that the product of
// subset elements is a factor of N
static int maximizeSubset(int N, int []arr,
                   int M, int x)
{
    // Base Case
    if (x == M) {
        return 0;
    }

    // Stores maximum size of valid subset
    int ans = 0;

    // Traverse the given array
    for (int i = x; i < M; i++) {

        // If N % arr[i] = 0, include arr[i]
        // in a subset and recursively call
        // for the remaining array integers
        if (N % arr[i] == 0) {
            ans = Math.Max(
                ans, maximizeSubset(
                         N / arr[i], arr,
                         M, x + 1)
                         + 1);
        }
    }

    // Return Answer
    return ans;
}

// Driver Code
public static void Main()
{
    int N = 64;
    int []arr = { 1, 2, 4, 8, 16, 32 };
    int M = arr.Length;

    Console.Write(maximizeSubset(N, arr, M,0));
}
}

// This code is contributed by ipg2016107.

Javascript

<script>
       // JavaScript Program to implement
       // the above approach

       // Function to find the maximum size of
       // the subset such that the product of
       // subset elements is a factor of N
       function maximizeSubset(N, arr,
           M, x = 0)
       {

           // Base Case
           if (x == M) {
               return 0;
           }

           // Stores maximum size of valid subset
           let ans = 0;

           // Traverse the given array
           for (let i = x; i < M; i++) {

               // If N % arr[i] = 0, include arr[i]
               // in a subset and recursively call
               // for the remaining array integers
               if (N % arr[i] == 0) {
                   ans = Math.max(
                       ans, maximizeSubset(
                           Math.floor(N / arr[i]), arr,
                           M, x + 1)
                   + 1);
               }
           }

           // Return Answer
           return ans;
       }

       // Driver Code
       let N = 64;
       let arr = [1, 2, 4, 8, 16, 32];
       let M = arr.length;

       document.write(maximizeSubset(N, arr, M));

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

Output:

4

Time Complexity: O(2^N)
Auxiliary Space: O(1)