Count of elements in given Array divisible by all elements in their prefix

Given an array arr[] containing N positive integers, the task is to find the total number of elements in the array that are divisible by all the elements present before them.

Examples:

Input: arr[] = {10, 6, 60, 120, 30, 360}
Output: 3
Explanation: 60, 120 and 360 are the required elements.

Input: arr[] = {2, 6, 5, 60}
Output: 2
Explanation: 6 and 60 are the elements.

Brute Force/ Naive Approach:

The brute force approach to solving the problem of Count of elements in given Array divisible by all elements in their prefix would be to check each element of the array one by one and calculate if it’s divisible by all its previous elements.

Steps to implement the approach:

• Initialize a variable ans to 0 to count the number of elements that satisfy the condition.
• Traverse the array from index 1 to N-1.
• For each index i, traverse the array from index 0 to i-1 to check if the element is divisible by all its previous elements.
• If the element is divisible by all its previous elements, increment the ans variable.
• Return the ans variable as the final count.

Below is the implementation of the approach:

C++

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

// Function to return total number of
// elements which are divisible by
// all their previous elements
int countElements(int arr[], int N)
{
    int ans = 0; // Initialize the count of elements to zero
    for (int i = 1; i < N;
         i++) { // Traverse the array from index 1 to N-1
        bool is_divisible
            = true; // Initialize the flag to true
        for (int j = 0; j < i; j++) { // Traverse the array
                                      // from index 0 to i-1
            if (arr[i] % arr[j]
                != 0) { // Check if the i-th element is not
                        // divisible by any of the previous
                        // elements
                is_divisible
                    = false; // Set the flag to false
                break; // Break out of the inner loop
            }
        }
        if (is_divisible) { // If the i-th element is
                            // divisible by all its previous
                            // elements
            ans++; // Increment the count of elements
        }
    }
    return ans; // Return the count of elements that are
                // divisible by all their previous elements
}

// Driver code
int main()
{
    int arr[] = {
        10, 6, 60, 120, 30, 360
    }; // Initialize the input array

    int N
        = sizeof(arr)
          / sizeof(
              int); // Calculate the size of the input array
    cout << countElements(
        arr, N); // Call the countElements() function and
                 // print the result
    return 0;
}

Java

import java.util.*;

public class Main {
    // Function to return total number of
    // elements which are divisible by
    // all their previous elements
    static int countElements(int[] arr, int N)
    {
        int ans
            = 0; // Initialize the count of elements to zero
        for (int i = 1; i < N; i++) { // Traverse the array
                                      // from index 1 to N-1
            boolean is_divisible
                = true; // Initialize the flag to true
            for (int j = 0; j < i;
                 j++) { // Traverse the array from index 0
                        // to i-1
                if (arr[i] % arr[j]
                    != 0) { // Check if the i-th element is
                            // not divisible by any of the
                            // previous elements
                    is_divisible
                        = false; // Set the flag to false
                    break; // Break out of the inner loop
                }
            }
            if (is_divisible) { // If the i-th element is
                                // divisible by all its
                                // previous elements
                ans++; // Increment the count of elements
            }
        }
        return ans; // Return the count of elements that are
                    // divisible by all their previous
                    // elements
    }

    // Driver code
    public static void main(String[] args)
    {
        int[] arr = {
            10, 6, 60, 120, 30, 360
        }; // Initialize the input array

        int N = arr.length; // Calculate the size of the
                            // input array
        System.out.println(countElements(
            arr, N)); // Call the countElements() function
                      // and print the result
    }
}

Python3

def count_elements(arr):
    ans = 0  # Initialize the count of elements to zero
    N = len(arr)  # Calculate the size of the input array
    for i in range(1, N):  # Traverse the array from index 1 to N-1
        is_divisible = True  # Initialize the flag to true
        for j in range(i):  # Traverse the array from index 0 to i-1
            if arr[i] % arr[j] != 0:  # Check if the i-th element is not divisible by any of the previous elements
                is_divisible = False  # Set the flag to false
                break  # Break out of the inner loop
        if is_divisible:  # If the i-th element is divisible by all its previous elements
            ans += 1  # Increment the count of elements
    return ans  # Return the count of elements that are divisible by all their previous elements

# Driver code
arr = [10, 6, 60, 120, 30, 360]  # Initialize the input array
print(count_elements(arr))  # Call the count_elements() function and print the result

C#

// C# program to count the number of elements which are
// divisible by all their previous elements.
using System;

class MainClass
{

  // Function to return total number of
  // elements which are divisible by
  // all their previous elements
  static int countElements(int[] arr, int N)
  {
    int ans
      = 0; // Initialize the count of elements to zero
    for (int i = 1; i < N; i++) { // Traverse the array
      // from index 1 to N-1
      bool is_divisible
        = true; // Initialize the flag to true
      for (int j = 0; j < i;
           j++) { // Traverse the array from index 0
        // to i-1
        if (arr[i] % arr[j]
            != 0) { // Check if the i-th element is
          // not divisible by any of the
          // previous elements
          is_divisible
            = false; // Set the flag to false
          break; // Break out of the inner loop
        }
      }
      if (is_divisible) { // If the i-th element is
        // divisible by all its
        // previous elements
        ans++; // Increment the count of elements
      }
    }
    return ans; // Return the count of elements that are
    // divisible by all their previous
    // elements
  }

  // Driver code
  public static void Main()
  {
    int[] arr = {
      10, 6, 60, 120, 30, 360
      }; // Initialize the input array

    int N = arr.Length; // Calculate the size of the
    // input array
    Console.WriteLine(countElements(
      arr, N)); // Call the countElements() function
    // and print the result
  }
}

Javascript

// Function to return total number of
// elements which are divisible by
// all their previous elements
function countElements(arr, N) {
    let ans = 0;// Initialize the count of elements to zero

    // Traverse the array
    // from index 1 to N-1
    for (let i = 1; i < N; i++) {
        let is_divisible = true; // Initialize the flag to true

            // Traverse the array from index 0
            // to i-1
            for (let j = 0; j < i; j++) {

                // Check if the i-th element is
                // not divisible by any of the
                // previous elements
                if (arr[i] % arr[j] !== 0) {
                    is_divisible = false; // Set the flag to false
                    break; // Break out of the inner loop
                }
            }

        // If the i-th element is
        // divisible by all its
        // previous elements
        if (is_divisible) {
        ans++;// Increment the count of elements
        }
    }
    // Return the count of elements that are
    // divisible by all their previous
    // elements
    return ans;
}

// Usage
let arr = [10, 6, 60, 120, 30, 360];
let N = arr.length;
console.log(countElements(arr, N)); // Output: count of elements that are divisible by all previous elements in the array

// This code is contributed by shivhack999

Output

3

Time Complexity: O(N² )
Space Complexity: O(1)

Another Approach: As known, that any number X is divided by {X₁, X₂, X₃, X₄, . . ., X_n}, if X is divided by LCM of {X₁, X₂, X₃, X₄, …, X_n). And LCM of any number A, B is [(A*B)/gcd(A, B)]. Now to solve this problem, follow the below steps:

1. Create a variable ans which stores the final answer and initializes it with 0.
2. Create another variable lcm which stores LCM up to the i^th element while iterating through the array. Initialize lcm with arr[0].
3. Iterate over the array from i = 1 to i = N, and in each iteration, check if arr[i] is divided by lcm. If yes increment ans by 1. Also, update lcm with lcm up to i^th element.
4. Print ans as the final answer to this problem.

Below is the implementation of the above approach:

C++

// C++ code to implement the above approach
#include <bits/stdc++.h>
using namespace std;

// Function to return total number of
// elements which are divisible by
// all their previous elements
int countElements(int arr[], int N)
{
    int ans = 0;
    int lcm = arr[0];
    for (int i = 1; i < N; i++) {

        // To check if number is divisible
        // by lcm of all previous elements
        if (arr[i] % lcm == 0) {
            ans++;
        }

        // Updating LCM
        lcm = (lcm * arr[i]) / __gcd(lcm, arr[i]);
    }
    return ans;
}

// Driver code
int main()
{
    int arr[] = { 10, 6, 60, 120, 30, 360 };

    int N = sizeof(arr) / sizeof(int);
    cout << countElements(arr, N);
    return 0;
}

Java

// Java program for the above approach
import java.util.*;
public class GFG {

  // Recursive function to return gcd of a and b
  static int __gcd(int a, int b)
  {

    // Everything divides 0
    if (a == 0)
      return b;
    if (b == 0)
      return a;

    // base case
    if (a == b)
      return a;

    // a is greater
    if (a > b)
      return __gcd(a - b, b);
    return __gcd(a, b - a);
  }

  // Function to return total number of
  // elements which are divisible by
  // all their previous elements
  static int countElements(int arr[], int N)
  {
    int ans = 0;
    int lcm = arr[0];
    for (int i = 1; i < N; i++) {

      // To check if number is divisible
      // by lcm of all previous elements
      if (arr[i] % lcm == 0) {
        ans++;
      }

      // Updating LCM
      lcm = (lcm * arr[i]) / __gcd(lcm, arr[i]);
    }
    return ans;
  }

  public static void main(String args[])
  {
    int arr[] = { 10, 6, 60, 120, 30, 360 };
    int N = arr.length;
    System.out.print(countElements(arr, N));
  }
}

// This code is contributed by Samim Hossain Mondal.

Python3

# Python code for the above approach

# Recursive function to return gcd of a and b
def __gcd(a, b):

    # Everything divides 0
    if (a == 0):
        return b;
    if (b == 0):
        return a;

    # base case
    if (a == b):
        return a;

    # a is greater
    if (a > b):
        return __gcd(a - b, b);
    return __gcd(a, b - a);

# Function to return total number of
# elements which are divisible by
# all their previous elements
def countElements(arr, N):
    ans = 0;
    lcm = arr[0];
    for i in range(1, N):

        # To check if number is divisible
        # by lcm of all previous elements
        if (arr[i] % lcm == 0):
            ans += 1

        # Updating LCM
        lcm = (lcm * arr[i]) / __gcd(lcm, arr[i]);
    return ans;

# Driver code
arr = [10, 6, 60, 120, 30, 360];

N = len(arr)
print(countElements(arr, N));

# This code is contributed by Saurabh Jaiswal

C#

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

public class GFG {

  // Recursive function to return gcd of a and b
  static int __gcd(int a, int b)
  {

    // Everything divides 0
    if (a == 0)
      return b;
    if (b == 0)
      return a;

    // base case
    if (a == b)
      return a;

    // a is greater
    if (a > b)
      return __gcd(a - b, b);
    return __gcd(a, b - a);
  }

  // Function to return total number of
  // elements which are divisible by
  // all their previous elements
  static int countElements(int[] arr, int N)
  {
    int ans = 0;
    int lcm = arr[0];
    for (int i = 1; i < N; i++) {

      // To check if number is divisible
      // by lcm of all previous elements
      if (arr[i] % lcm == 0) {
        ans++;
      }

      // Updating LCM
      lcm = (lcm * arr[i]) / __gcd(lcm, arr[i]);
    }
    return ans;
  }

  public static void Main()
  {
    int[] arr = { 10, 6, 60, 120, 30, 360 };
    int N = arr.Length;
    Console.Write(countElements(arr, N));
  }
}

// This code is contributed by ukasp.

Javascript

<script>
      // JavaScript code for the above approach

      // Recursive function to return gcd of a and b
      function __gcd(a, b) {
          // Everything divides 0
          if (a == 0)
              return b;
          if (b == 0)
              return a;

          // base case
          if (a == b)
              return a;

          // a is greater
          if (a > b)
              return __gcd(a - b, b);
          return __gcd(a, b - a);
      }


      // Function to return total number of
      // elements which are divisible by
      // all their previous elements
      function countElements(arr, N) {
          let ans = 0;
          let lcm = arr[0];
          for (let i = 1; i < N; i++) {

              // To check if number is divisible
              // by lcm of all previous elements
              if (arr[i] % lcm == 0) {
                  ans++;
              }

              // Updating LCM
              lcm = (lcm * arr[i]) / __gcd(lcm, arr[i]);
          }
          return ans;
      }

      // Driver code

      let arr = [10, 6, 60, 120, 30, 360];

      let N = arr.length
      document.write(countElements(arr, N));

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

Output

3

Time complexity: O(N * logD) where D is the maximum array element
Auxiliary Space: O(N)