Queries to count numbers from given range which are divisible by all its digits

Given a 2D array arr[][] with each row of the form of a query { L, R }, the task is to count the numbers in the range [L, R] such that the number is divisible by all of its non-zero digit.

Examples:

Input: arr[][] ={ {1, 5}, {12, 14} } 
Output: 5 1 
Explanation: 
Query1: All the numbers in the range [1, 5] are divisible by their digits. 
Query2: Numbers in the range [12, 14] which are divisible by all of its digits is 12 only.

Input: arr[][] = { {1, 20} } 
Output:14 
Explanation: 
Numbers in the range [1, 20] which are divisible by all of its non-zero digits are: { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20}

Naive Approach: The simplest approach to solve this problem is to traverse the array arr[][] and for every i^th row of the array, iterate over the range [L, R]. For every element in the range, check if the number is divisible by all of its non-zero digits or not. If found to be true, then increment the count. Finally, print the count obtained. 
Time Complexity: O(N * (R - L)), where N is the count of rows 
Auxiliary Space: O(1)

Efficient approach: The above approach can be optimized by finding the maximum possible value of arr[i][1] and precompute the count of numbers which is divisible by its non-zero digits using Prefix Sum technique. Follow the steps below to solve the problem:

• Initialize a variable, say Max, to store the maximum possible value of arr[i][1].
• Initialize an array, say prefCntDiv[], where prefCntDiv[i] stores the count of numbers from the range [1, i] which is divisible by its non-zero digits.
• Iterate over the range [1, Max]. For every i^th iteration, check if i is divisible by all of its non-zero digits or not. If found to be true, then update prefCntDiv[i] = prefCntDiv[i - 1] + 1.
• Otherwise, update prefCntDiv[i] = prefCntDiv[i - 1].
• Traverse the array arr[]. For every i^th row of the array, print prefCntDiv[arr[i][1]] - prefCntDiv[arr[i][0] - 1].

Below is the implementation of the above approach:

// C++ program to implement
// the above approach

#include <bits/stdc++.h>
using namespace std;
#define Max 1000005

// Function to check if a number is divisible
// by all of its non-zero digits or not
bool CheckDivByAllDigits(int number)
{
    // Stores the number
    int n = number;

    // Iterate over the digits
    // of the numbers
    while (n > 0) {

        // If digit of number
        // is non-zero
        if (n % 10)

            // If number is not divisible
            // by its current digit
            if (number % (n % 10)) {

                return false;
            }

        // Update n
        n /= 10;
    }
    return true;
}

// Function to count of numbers which are
// divisible by all of its non-zero
// digits in the range [1, i]
void cntNumInRang(int arr[][2], int N)
{

    // Stores count of numbers which are
    // divisible by all of its non-zero
    // digits in the range [1, i]
    int prefCntDiv[Max] = { 0 };

    // Iterate over the range [1, Max]
    for (int i = 1; i <= Max; i++) {

        // Update
        prefCntDiv[i] = prefCntDiv[i - 1]
                        + (CheckDivByAllDigits(i));
    }

    // Traverse the array, arr[]
    for (int i = 0; i < N; i++)
        cout << (prefCntDiv[arr[i][1]]
                 - prefCntDiv[arr[i][0] - 1])
             << " ";
}

// Driver Code
int main()
{
    int arr[][2] = { { 1, 5 }, { 12, 14 } };

    int N = sizeof(arr) / sizeof(arr[0]);
    cntNumInRang(arr, N);
    return 0;
}

// Java program to implement
// the above approach
import java.io.*;
class GFG
{
  public static int Max = 1000005;

  // Function to check if a number is divisible
  // by all of its non-zero digits or not
  static boolean CheckDivByAllDigits(int number)
  {

    // Stores the number
    int n = number;

    // Iterate over the digits
    // of the numbers
    while (n > 0)
    {

      // If digit of number
      // is non-zero
      if (n % 10 != 0)

        // If number is not divisible
        // by its current digit
        if (number % (n % 10) != 0) 
        {
          return false;
        }

      // Update n
      n /= 10;
    }
    return true;
  }

  // Function to count of numbers which are
  // divisible by all of its non-zero
  // digits in the range [1, i]
  static void cntNumInRang(int arr[][], int N)
  {

    // Stores count of numbers which are
    // divisible by all of its non-zero
    // digits in the range [1, i]
    int prefCntDiv[] = new int[Max + 1];

    // Iterate over the range [1, Max]
    for (int i = 1; i <= Max; i++)
    {

      int ans = 0;
      if (CheckDivByAllDigits(i))
        ans = 1;

      // Update
      prefCntDiv[i] = prefCntDiv[i - 1] + ans;
    }

    // Traverse the array, arr[]
    for (int i = 0; i < N; i++)
      System.out.print((prefCntDiv[arr[i][1]]
                        - prefCntDiv[arr[i][0] - 1])
                       + " ");
  }

  // Driver Code
  public static void main(String[] args)
  {
    int arr[][] = { { 1, 5 }, { 12, 14 } };
    int N = arr.length;
    cntNumInRang(arr, N);
  }
}

// This code is contributed by Dharanendra L V

# Python3 program to implement
# the above approach

# Function to check if a number is divisible
# by all of its non-zero digits or not
def CheckDivByAllDigits(number):
    
    # Stores the number
    n = number

    # Iterate over the digits
    # of the numbers
    while (n > 0):

        # If digit of number
        # is non-zero
        if (n % 10):

            # If number is not divisible
            # by its current digit
            if (number % (n % 10)):
                return False

        # Update n
        n //= 10
    return True

# Function to count of numbers which are
# divisible by all of its non-zero
# digits in the range [1, i]
def cntNumInRang(arr, N):
    global Max

    # Stores count of numbers which are
    # divisible by all of its non-zero
    # digits in the range [1, i]
    prefCntDiv = [0]*Max

    # Iterate over the range [1, Max]
    for i in range(1, Max):

        # Update
        prefCntDiv[i] = prefCntDiv[i - 1] + (CheckDivByAllDigits(i))

    # Traverse the array, arr[]
    for i in range(N):
        print(prefCntDiv[arr[i][1]]- prefCntDiv[arr[i][0] - 1], end = " ")

# Driver Code
if __name__ == '__main__':
    arr =[ [ 1, 5 ], [12, 14]]
    Max = 1000005

    N = len(arr)
    cntNumInRang(arr, N)

    # This code is contributed by mohit kumar 29.

// C# program to implement
// the above approach
using System;
class GFG
{
  public static int Max = 1000005;

  // Function to check if a number is divisible
  // by all of its non-zero digits or not
  static bool CheckDivByAllDigits(int number)
  {

    // Stores the number
    int n = number;

    // Iterate over the digits
    // of the numbers
    while (n > 0) {

      // If digit of number
      // is non-zero
      if (n % 10 != 0)

        // If number is not divisible
        // by its current digit
        if (number % (n % 10) != 0)
        {
          return false;
        }

      // Update n
      n /= 10;
    }
    return true;
  }

  // Function to count of numbers which are
  // divisible by all of its non-zero
  // digits in the range [1, i]
  static void cntNumInRang(int[, ] arr, int N)
  {

    // Stores count of numbers which are
    // divisible by all of its non-zero
    // digits in the range [1, i]
    int[] prefCntDiv = new int[Max + 1];

    // Iterate over the range [1, Max]
    for (int i = 1; i <= Max; i++) {

      int ans = 0;
      if (CheckDivByAllDigits(i))
        ans = 1;

      // Update
      prefCntDiv[i] = prefCntDiv[i - 1] + ans;
    }

    // Traverse the array, arr[]
    for (int i = 0; i < N; i++)
      Console.Write((prefCntDiv[arr[i, 1]]
                     - prefCntDiv[arr[i, 0] - 1])
                    + " ");
  }

  // Driver Code
  public static void Main(string[] args)
  {
    int[, ] arr = { { 1, 5 }, { 12, 14 } };
    int N = arr.GetLength(0);
    cntNumInRang(arr, N);
  }
}

// This code is contributed by chitranayal.

<script>
// Javascript program to implement
// the above approach

const Max = 1000005;

// Function to check if a number is divisible
// by all of its non-zero digits or not
function CheckDivByAllDigits(number)
{
    // Stores the number
    let n = number;

    // Iterate over the digits
    // of the numbers
    while (n > 0) {

        // If digit of number
        // is non-zero
        if (n % 10)

            // If number is not divisible
            // by its current digit
            if (number % (n % 10)) {

                return false;
            }

        // Update n
        n = parseInt(n / 10);
    }
    return true;
}

// Function to count of numbers which are
// divisible by all of its non-zero
// digits in the range [1, i]
function cntNumInRang(arr, N)
{

    // Stores count of numbers which are
    // divisible by all of its non-zero
    // digits in the range [1, i]
    let prefCntDiv = new Array(Max).fill(0);

    // Iterate over the range [1, Max]
    for (let i = 1; i <= Max; i++) {

        // Update
        prefCntDiv[i] = prefCntDiv[i - 1]
                        + (CheckDivByAllDigits(i));
    }

    // Traverse the array, arr[]
    for (let i = 0; i < N; i++)
        document.write((prefCntDiv[arr[i][1]]
                 - prefCntDiv[arr[i][0] - 1])
             + " ");
}

// Driver Code
    let arr = [ [ 1, 5 ], [ 12, 14 ] ];

    let N = arr.length;
    cntNumInRang(arr, N);

</script>

5 1

Time Complexity: O(N + Max), where Max is the maximum value of arr[i][1]
Auxiliary Space: O(N)

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Article Tags :

• Greedy
• Searching
• Mathematical
• Technical Scripter
• Competitive Programming
• DSA
• divisibility
• array-range-queries
• prefix-sum
• number-digits

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