Count divisors of array multiplication

Given an array with N elements, the task is to find the count of factors of a number X, which is the product of all array elements.

Examples:

Input : 5 5
Output : 3
5 * 5 = 25, the factors of 25 are 1, 5, 25
whose count is 3
 
Input : 3 5 7
Output : 8
3 * 5 * 7 = 105, the factors of 105 are 1,
3, 5, 7, 15, 21, 35, 105 whose count is 8

Method 1 (Simple but causes overflow):

1. Multiply all the elements of the array. 
2. Count divisors in the number obtained after multiplication. 

Implementation:

C++

// A simple C++ program to count divisors
// in array multiplication.
#include <iostream>
using namespace std;

// To count number of factors in a number
int counDivisors(int X)
{
    // Initialize count with 0
    int count = 0;
    // Increment count for every factor
    // of the given number X.
    for (int i = 1; i <= X; ++i) {
        if (X % i == 0) {
            count++;
        }
    }

    // Return number of factors
    return count;
}

// Returns number of divisors in array
// multiplication
int countDivisorsMult(int arr[], int n)
{
    // Multiplying all elements of
    // the given array.
    int mul = 1;
    for (int i = 0; i < n; ++i)
        mul *= arr[i];

    // Calling function which count number of factors
    // of the number
    return counDivisors(mul);
}

// Driver code
int main()
{
    int arr[] = { 2, 4, 6 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << countDivisorsMult(arr, n) << endl;
    return 0;
}

Java

// A simple Java program to count divisors
// in array multiplication.

class GFG
{
    // To count number of factors in a number
    static int counDivisors(int X)
    {
        // Initialize count with 0
        int count = 0;

        // Increment count for every factor
        // of the given number X.
        for (int i = 1; i <= X; ++i)
        {
            if (X % i == 0) {
                count++;
            }
        }

        // Return number of factors
        return count;
    }

    // Returns number of divisors in array
    // multiplication
    static int countDivisorsMult(int arr[], int n)
    {
        // Multiplying all elements of
        // the given array.
        int mul = 1;
        for (int i = 0; i < n; ++i)
            mul *= arr[i];

        // Calling function which count
        // number of factors of the number
        return counDivisors(mul);
    }


    // Driver code
    public static void main (String[] args)
    {
        int arr[] = { 2, 4, 6 };
        int n = arr.length;
        System.out.println(countDivisorsMult(arr, n));
    }
}

// This code is contributed by Anant Agarwal.

Python3

# A simple Python program
# to count divisors
# in array multiplication.

# To count number of
# factors in a number
def counDivisors(X):

    # Initialize count with 0
    count = 0
    # Increment count for
    # every factor
    # of the given number X.
    for i in range(1, X + 1):
        if (X % i == 0):
            count += 1

    # Return number of factors
    return count

# Returns number of
# divisors in array
# multiplication
def countDivisorsMult(arr, n):

    # Multiplying all elements of
    # the given array.
    mul = 1
    for i in range(n):
        mul *= arr[i]

    # Calling function which
    # count number of factors
    # of the number
    return counDivisors(mul)

# Driver code

arr = [ 2, 4, 6 ]
n =len(arr)

print(countDivisorsMult(arr, n))

# This code is contributed
# by Anant Agarwal.

C#

// C# program to count divisors
// in array multiplication.
using System;

class GFG {

    // To count number of factors
    // in a number
    static int counDivisors(int X)
    {

        // Initialize count with 0
        int count = 0;

        // Increment count for every
        // factor of the given
        // number X.
        for (int i = 1; i <= X; ++i)
        {
            if (X % i == 0) {
                count++;
            }
        }

        // Return number of factors
        return count;
    }

    // Returns number of divisors in
    // array multiplication
    static int countDivisorsMult(
                    int []arr, int n)
    {

        // Multiplying all elements
        // of the given array.
        int mul = 1;

        for (int i = 0; i < n; ++i)
            mul *= arr[i];

        // Calling function which
        // count number of factors
        // of the number
        return counDivisors(mul);
    }

    // Driver code
    public static void Main ()
    {

        int []arr = { 2, 4, 6 };
        int n = arr.Length;

        Console.Write(
         countDivisorsMult(arr, n));
    }
}

// This code is contributed by
// nitin mittal.

PHP

<?php
// A simple PHP program to count divisors
// in array multiplication.

// To count number of factors in a number
function counDivisors($X)
{

    // Initialize count with 0
    $count = 0;

    // Increment count for every factor
    // of the given number X.
    for ($i = 1; $i <= $X; ++$i) {
        if ($X % $i == 0) {
            $count++;
        }
    }

    // Return number of factors
    return $count;
}

// Returns number of divisors in array
// multiplication
function countDivisorsMult($arr, $n)
{

    // Multiplying all elements of
    // the given array.
    $mul = 1;
    for ($i = 0; $i < $n; ++$i)
        $mul *= $arr[$i];

    // Calling function which count
    // number of factors of the number
    return counDivisors($mul);
}

// Driver code
$arr = array(2, 4, 6);
$n = sizeof($arr);
echo countDivisorsMult($arr, $n);

// This code is contributed by nitin mittal
?>

Javascript

<script>
// Javascript program to count divisors
// in array multiplication.

    // To count number of factors in a number
    function counDivisors(X)
    {
        // Initialize count with 0
        let count = 0;

        // Increment count for every factor
        // of the given number X.
        for (let i = 1; i <= X; ++i)
        {
            if (X % i == 0) {
                count++;
            }
        }

        // Return number of factors
        return count;
    }

    // Returns number of divisors in array
    // multiplication
   function countDivisorsMult(arr, n)
    {
        // Multiplying all elements of
        // the given array.
        let mul = 1;
        for (let i = 0; i < n; ++i)
            mul *= arr[i];

        // Calling function which count
        // number of factors of the number
        return counDivisors(mul);
    }

// driver function
        let arr = [ 2, 4, 6 ];
        let n = arr.length;
        document.write(countDivisorsMult(arr, n));

  // This code is contributed by code_hunt.
</script>

Output

10

Time Complexity: O(m), m is the multiplication of all elements of array 
Auxiliary Space: O(1)

Method 2 (Avoids overflow):

1. Find a maximum element in the array 
2. Find prime numbers smaller than the maximum element 
3. Find the number of overall occurrences of each prime factor in the whole array by traversing all array elements and finding their prime factors. We use hashing to count occurrences. 
4. Let the counts of occurrences of prime factors be a1, a2, …aK, if we have K distinct prime factors, then the answer will be: (a1+1)(a2+1)(…)*(aK+1).

Implementation:

C++

// C++ program to count divisors in array multiplication.
#include <bits/stdc++.h>
using namespace std;

void SieveOfEratosthenes(int largest, vector<int> &prime)
{

    // Create a boolean array "isPrime[0..n]" and initialize
    // all entries it as true. A value in isPrime[i] will
    // finally be false if i is Not a isPrime, else true.
    bool isPrime[largest+1];
    memset(isPrime, true, sizeof(isPrime));

    for (int p = 2; p * p <= largest; p++)
    {

        // If isPrime[p] is not changed, then it is a isPrime
        if (isPrime[p] == true)
        {

            // Update all multiples of p
            for (int i = p * 2; i <= largest; i += p)
                isPrime[i] = false;
        }
    }

    // Print all isPrime numbers
    for (int p = 2; p <= largest; p++)
        if (isPrime[p])
            prime.push_back(p);
}

// Returns number of divisors in array
// multiplication
int countDivisorsMult(int arr[], int n)
{

    // Find all prime numbers smaller than
    // the largest element.
    int largest = *max_element(arr, arr+n);
    vector<int> prime;
    SieveOfEratosthenes(largest, prime);

    // Find counts of occurrences of each prime
    // factor
    unordered_map<int, int> mp;
    for (int i = 0; i < n; i++)
    {
        for (int j = 0; j < prime.size(); j++)
        {
            while(arr[i] > 1 && arr[i]%prime[j] == 0)
            {
                arr[i] /= prime[j];
                mp[prime[j]]++;
            }
        }
        if (arr[i] != 1)
            mp[arr[i]]++;
    }

    // Compute count of all divisors using counts
    // prime factors.
    long long int res = 1;
    for (auto it : mp)
       res *= (it.second + 1L);
    return res;
}

// Driver code
int main()
{
    int arr[] = { 2, 4, 6 };
    int n = sizeof(arr) / sizeof(arr[0]);
    cout << countDivisorsMult(arr, n) << endl;
    return 0;
}

Java

// Java program to count divisors in array multiplication.
import java.io.*;
import java.util.*;
class GFG
{

  static void SieveOfEratosthenes(int largest, ArrayList<Integer> prime)
  {

    // Create a boolean array "isPrime[0..n]" and initialize
    // all entries it as true. A value in isPrime[i] will
    // finally be false if i is Not a isPrime, else true.
    boolean[] isPrime = new boolean[largest + 1];
    Arrays.fill(isPrime, true);

    for (int p = 2; p * p <= largest; p++)
    {

      // If isPrime[p] is not changed, then it is a isPrime
      if (isPrime[p] == true)
      {

        // Update all multiples of p
        for (int i = p * 2; i <= largest; i += p)
          isPrime[i] = false;
      }
    }

    // Print all isPrime numbers
    for (int p = 2; p <= largest; p++)
      if (isPrime[p])
        prime.add(p);
  }

  // Returns number of divisors in array
  // multiplication
  static long countDivisorsMult(int[] arr, int n)
  {

    // Find all prime numbers smaller than
    // the largest element.
    int largest = 0;
    for(int a : arr )
    {
      largest=Math.max(largest, a);
    }

    ArrayList<Integer> prime = new ArrayList<Integer>();
    SieveOfEratosthenes(largest, prime);

    // Find counts of occurrences of each prime
    // factor
    Map<Integer,Integer> mp = new HashMap<>();
    for (int i = 0; i < n; i++)
    {
      for (int j = 0; j < prime.size(); j++)
      {
        while(arr[i] > 1 && arr[i]%prime.get(j) == 0)
        {
          arr[i] /= prime.get(j);
          if(mp.containsKey(prime.get(j)))
          {
            mp.put(prime.get(j), mp.get(prime.get(j)) + 1);
          }
          else
          {
            mp.put(prime.get(j), 1);
          }
        }
      }
      if (arr[i] != 1)
      {
        if(mp.containsKey(arr[i]))
        {
          mp.put(arr[i], mp.get(arr[i]) + 1);
        }
        else
        {
          mp.put(arr[i], 1);
        }
      }
    }

    // Compute count of all divisors using counts
    // prime factors.
    long res = 1;
    for (int it : mp.keySet())
      res *= (mp.get(it) + 1L);
    return res;
  }

  // Driver code
  public static void main (String[] args) {
    int arr[] = { 2, 4, 6 };
    int n = arr.length;
    System.out.println(countDivisorsMult(arr, n));
  }
}

// This code is contributed by avanitrachhadiya2155

Python3

# Python 3 program to count divisors in array multiplication.
from collections import defaultdict
def SieveOfEratosthenes(largest, prime):

    # Create a boolean array "isPrime[0..n]" and initialize
    # all entries it as true. A value in isPrime[i] will
    # finally be false if i is Not a isPrime, else true.
    isPrime = [True] * (largest + 1)

    p = 2
    while p * p <= largest:

        # If isPrime[p] is not changed, then it is a isPrime
        if (isPrime[p] == True):

            # Update all multiples of p
            for i in range(p * 2, largest + 1, p):
                isPrime[i] = False
        p += 1

    # Print all isPrime numbers
    for p in range(2, largest + 1):
        if (isPrime[p]):
            prime.append(p)

# Returns number of divisors in array
# multiplication
def countDivisorsMult(arr, n):

    # Find all prime numbers smaller than
    # the largest element.
    largest = max(arr)
    prime = []
    SieveOfEratosthenes(largest, prime)

    # Find counts of occurrences of each prime
    # factor
    mp = defaultdict(int)
    for i in range(n):
        for j in range(len(prime)):
            while(arr[i] > 1 and arr[i] % prime[j] == 0):
                arr[i] //= prime[j]
                mp[prime[j]] += 1
        if (arr[i] != 1):
            mp[arr[i]] += 1

    # Compute count of all divisors using counts
    # prime factors.
    res = 1
    for it in mp.values():
        res *= (it + 1)
    return res

# Driver code
if __name__ == "__main__":

    arr = [2, 4, 6]
    n = len(arr)
    print(countDivisorsMult(arr, n))

    # This code is contributed by chitranayal.

C#

// C# program to count divisors in array multiplication.
using System;
using System.Collections.Generic;

class GFG{

static void SieveOfEratosthenes(int largest,
                                List<int> prime)
{

    // Create a boolean array "isPrime[0..n]" and initialize
    // all entries it as true. A value in isPrime[i] will
    // finally be false if i is Not a isPrime, else true.
    bool[] isPrime = new bool[largest + 1];
    Array.Fill(isPrime, true);

    for(int p = 2; p * p <= largest; p++)
    {

        // If isPrime[p] is not changed, then it is a isPrime
        if (isPrime[p] == true)
        {

            // Update all multiples of p
            for (int i = p * 2; i <= largest; i += p)
                isPrime[i] = false;
        }
    }

    // Print all isPrime numbers
    for(int p = 2; p <= largest; p++)
        if (isPrime[p])
            prime.Add(p);
}

// Returns number of divisors in array
// multiplication
static long countDivisorsMult(int[] arr, int n)
{

    // Find all prime numbers smaller than
    // the largest element.
    int largest = 0;
    foreach(int a in arr )
    {
        largest = Math.Max(largest, a);
    }

    List<int> prime = new List<int>();
    SieveOfEratosthenes(largest, prime);

    // Find counts of occurrences of each prime
    // factor
    Dictionary<int,
               int> mp = new Dictionary<int,
                                        int>();
    for(int i = 0; i < n; i++)
    {
        for(int j = 0; j < prime.Count; j++)
        {
            while(arr[i] > 1 && arr[i] % prime[j] == 0)
            {
                arr[i] /= prime[j];
                if (mp.ContainsKey(prime[j]))
                {
                    mp[prime[j]]++;
                }
                else
                {
                    mp.Add(prime[j], 1);
                }
            }
        }
        if (arr[i] != 1)
        {
            if(mp.ContainsKey(arr[i]))
            {
                mp[arr[i]]++;
            }
            else
            {
                mp.Add(arr[i], 1);
            }
        }
    }

    // Compute count of all divisors using counts
    // prime factors.
    long res = 1;
    foreach(KeyValuePair<int, int> it in mp)
        res *= (it.Value + 1L);

    return res;
}

// Driver code
static public void Main()
{
    int[] arr = { 2, 4, 6 };
    int n = arr.Length;

    Console.WriteLine(countDivisorsMult(arr, n));
}
}

// This code is contributed by rag2127

Javascript

<script>

// javascript program to count divisors in array multiplication.

function SieveOfEratosthenes(largest, prime)
{

    // Create a boolean array "isPrime[0..n]" and initialize
    // all entries it as true. A value in isPrime[i] will
    // finally be false if i is Not a isPrime, else true.
    var isPrime = Array(largest+1).fill(true);

    var p,i;
    for (p = 2; p * p <= largest; p++)
    {

        // If isPrime[p] is not changed, then it is a isPrime
        if (isPrime[p] == true)
        {

            // Update all multiples of p
            for(i = p * 2; i <= largest; i += p)
                isPrime[i] = false;
        }
    }

    // Print all isPrime numbers
    for (p = 2; p <= largest; p++)
        if (isPrime[p])
            prime.push(p);
}

// Returns number of divisors in array
// multiplication
function countDivisorsMult(arr, n)
{

    // Find all prime numbers smaller than
    // the largest element.
    var largest = Math.max.apply(null,arr);
    var prime = [];
    SieveOfEratosthenes(largest, prime);

    // Find counts of occurrences of each prime
    // factor
    var j;
    var mp = new Map();
    for (i = 0; i < n; i++)
    {
        for (j = 0; j < prime.length; j++)
        {
            while(arr[i] > 1 && arr[i]%prime[j] == 0)
            {
                arr[i] /= prime[j];
                if(mp.has(prime[j]))
                 mp.set(prime[j],mp.get(prime[j])+1);
                else
                  mp.set(prime[j],1);
            }
        }
        if (arr[i] != 1){
            if(mp.has(arr[i]))
                 mp.set(arr[i],mp.get(arr[i])+1);
                else
                  mp.set(arr[i],1);
        }
    }

    // Compute count of all divisors using counts
    // prime factors.
    var res = 1;
    for (const [key, value] of mp.entries()) {
        res *= (value + 1);
    }

    return res;
}

// Driver code
    var arr =  [2, 4, 6];
    var n = arr.length;
    document.write(countDivisorsMult(arr, n));

</script>

Output

10

Time Complexity: O(n²) 
Auxiliary Space: O(n)