Count all prefixes of the given binary array which are divisible by x

Given a binary array arr[] and an integer x, the task is to count all the prefixes of the given array which are divisible by x.
Note: The ith prefix from arr[0] to arr[i] is interpreted as a binary number (from most-significant-bit to least-significant-bit.)

Examples:

Input: arr[] = {0, 1, 0, 1, 1}, x = 5
Output: 2
0 = 0
01 = 1
010 = 2
0101 = 5
01011 = 11
0 and 0101 are the only prefixes divisible by 5.

Input: arr[] = {1, 0, 1, 0, 1, 1, 0}, x = 2
Output: 3

Naive Approach: Iterate from 0 to i to convert each binary prefix to decimal and check whether the number is divisible by x or not.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the count of total
// binary prefix which are divisible by x
int CntDivbyX(int arr[], int n, int x)
{
  
    // Initialize with zero
    int number = 0;
    int count = 0;
  
    for (int i = 0; i < n; i++) {
  
        // Convert all prefixes to decimal
        number = number * 2 + arr[i];
  
        // If number is divisible by x
        // then increase count
        if ((number % x == 0))
            count += 1;
    }
  
    return count;
}
  
// Driver code
int main()
{
    int arr[] = { 1, 0, 1, 0, 1, 1, 0 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int x = 2;
    cout << CntDivbyX(arr, n, x);
  
    return 0;
}

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Java

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// Java implementation of the approach 
class GfG
{
  
    // Function to return the count of total 
    // binary prefix which are divisible by x 
    static int CntDivbyX(int arr[], int n, int x) 
    
      
        // Initialize with zero 
        int number = 0
        int count = 0
      
        for (int i = 0; i < n; i++) 
        
      
            // Convert all prefixes to decimal 
            number = number * 2 + arr[i]; 
      
            // If number is divisible by x 
            // then increase count 
            if ((number % x == 0)) 
                count += 1
        
      
        return count; 
    
  
    // Driver Code
    public static void main(String []args)
    {
          
        int arr[] = { 1, 0, 1, 0, 1, 1, 0 }; 
        int n = arr.length; 
        int x = 2
        System.out.println(CntDivbyX(arr, n, x));
    }
}
  
// This code is contributed by Rituraj Jain

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Python3

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# Python 3 implementation of the approach
  
# Function to return the count of total
# binary prefix which are divisible by x
def CntDivbyX(arr, n, x):
      
    # Initialize with zero
    number = 0
    count = 0
  
    for i in range(n):
          
        # Convert all prefixes to decimal
        number = number * 2 + arr[i]
  
        # If number is divisible by x
        # then increase count
        if ((number % x == 0)):
            count += 1
  
    return count
  
# Driver code
if __name__ == '__main__':
    arr = [1, 0, 1, 0, 1, 1, 0]
    n = len(arr)
    x = 2
    print(CntDivbyX(arr, n, x))
  
# This code is contributed by
# Surendra_Gangwar

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C#

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// C# implementation of the approach 
using System;
  
class GfG
{
  
    // Function to return the count of total 
    // binary prefix which are divisible by x 
    static int CntDivbyX(int[] arr, int n, int x) 
    
      
        // Initialize with zero 
        int number = 0; 
        int count = 0; 
      
        for (int i = 0; i < n; i++) 
        
      
            // Convert all prefixes to decimal 
            number = number * 2 + arr[i]; 
      
            // If number is divisible by x 
            // then increase count 
            if ((number % x == 0)) 
                count += 1; 
        
      
        return count; 
    
  
    // Driver Code
    public static void Main()
    {
          
        int[] arr = { 1, 0, 1, 0, 1, 1, 0 }; 
        int n = arr.Length; 
        int x = 2; 
        Console.WriteLine(CntDivbyX(arr, n, x));
    }
}
  
// This code is contributed by Code_Mech.

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PHP

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<?php
// PHP implementation of the approach
  
// Function to return the count of total
// binary prefix which are divisible by x
function CntDivbyX($arr, $n, $x)
{
  
    // Initialize with zero
    $number = 0;
    $count = 0;
  
    for ($i = 0; $i < $n; $i++)
    {
  
        // Convert all prefixes to decimal
        $number = $number * 2 + $arr[$i];
  
        // If number is divisible by x
        // then increase count
        if (($number % $x == 0))
            $count += 1;
    }
  
    return $count;
}
  
// Driver code
$arr = array(1, 0, 1, 0, 1, 1, 0);
$n = sizeof($arr);
$x = 2;
echo CntDivbyX($arr, $n, $x);
  
// This code is contributed by Akanksha Rai

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Output:

3

Efficient Approach: As we see in the above approach we convert each binary prefix to decimal number like 0, 01, 010, 0101…. but as the value of n(size of array) increases then the resultant number will be very large and no. will be out of range of data type so we can make use of the modular properties.
Instead of doing number = number * 2 + arr[ i ] , we can do better as number = (number * 2 + arr[ i ] ) % x
Explanation: We start with number = 0 and repeatedly do number = number * 2 + arr[ i ] then in each iteration we’ll get a new term of the above sequence.

A = {1, 0, 1, 0, 1, 1, 0}
“1” = 0*2 + 1 = 1
“10” = 1*2 + 0 = 2
“101” = 2*2 + 1 = 5
“1010” = 5*2 + 0 = 10
“10101” = 10*2 + 1 = 21
“101011” = 21*2 + 1 = 43
“1010110” = 43*2 + 0 =86

Since we are repeatedly taking the remainders of the number at each step, at each step we have, newNum = oldNum * 2 + arr[i] .By the rules of modular arithmetic (a * b + c) % m is same as ((a * b) % m + c % m) % m. So, it doesn’t matter whether oldNum is the remainder or the original number, the answer would be correct.
Note: Similar article discussed here.

Below is the implementation of the above approach:

C++

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// C++ implementation of the approach
#include <bits/stdc++.h>
using namespace std;
  
// Function to return the count of total
// binary prefix which are divisible by x
int CntDivbyX(int arr[], int n, int x)
{
  
    // Initialize with zero
    int number = 0;
    int count = 0;
  
    for (int i = 0; i < n; i++) {
  
        // Instead of converting all prefixes
        // to decimal, take reminder with x
        number = (number * 2 + arr[i]) % x;
  
        // If number is divisible by x
        // then reminder = 0
        if (number == 0)
            count += 1;
    }
  
    return count;
}
  
// Driver code
int main()
{
  
    int arr[] = { 1, 0, 1, 0, 1, 1, 0 };
    int n = sizeof(arr) / sizeof(arr[0]);
    int x = 2;
    cout << CntDivbyX(arr, n, x);
  
    return 0;
}

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Java

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// Java implementation of the approach
class GFG {
  
    // Function to return the count of total
    // binary prefix which are divisible by x
    public static int CntDivbyX(int arr[], int n, int x)
    {
  
        // Initialize with zero
        int number = 0;
        int count = 0;
  
        for (int i = 0; i < n; i++) {
  
            // Instead of converting all prefixes
            // to decimal, take reminder with x
            number = (number * 2 + arr[i]) % x;
  
            // If number is divisible by x
            // then reminder = 0
            if (number == 0)
                count += 1;
        }
  
        return count;
    }
  
    // Driver code
    public static void main(String[] args)
    {
        int arr[] = { 1, 0, 1, 0, 1, 1, 0 };
        int n = 7;
        int x = 2;
        System.out.print(CntDivbyX(arr, n, x));
    }
}

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Python3

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# Python3 implementation of the approach
  
# Function to return the count of total
# binary prefix which are divisible by x
def CntDivbyX(arr, n, x):
  
    number = 0
  
    count = 0
  
    for i in range (0, n):
          
        # Instead of converting all prefixes 
        # to decimal, take reminder with x
        number = ( number * 2 + arr[i] ) % x
      
        # If number is divisible by x 
        # then reminder = 0
        if number == 0:
            count += 1
      
    return count
  
# Driver code
arr = [1, 0, 1, 0, 1, 1, 0]
n = 7
x = 2
print( CntDivbyX(arr, n, x) )

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C#

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// C# implementation of the approach
using System;
  
class GFG 
{
  
    // Function to return the count of total
    // binary prefix which are divisible by x
    public static int CntDivbyX(int []arr, int n, int x)
    {
  
        // Initialize with zero
        int number = 0;
        int count = 0;
  
        for (int i = 0; i < n; i++) 
        {
  
            // Instead of converting all prefixes
            // to decimal, take reminder with x
            number = (number * 2 + arr[i]) % x;
  
            // If number is divisible by x
            // then reminder = 0
            if (number == 0)
                count += 1;
        }
  
        return count;
    }
  
    // Driver code
    public static void Main()
    {
        int []arr = { 1, 0, 1, 0, 1, 1, 0 };
        int n = 7;
        int x = 2;
          
        Console.Write(CntDivbyX(arr, n, x));
    }
}
  
// This code is contributed by Ryuga

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PHP

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<?php
// PHP implementation of the approach
  
// Function to return the count of total
// binary prefix which are divisible by x
function CntDivbyX($arr, $n, $x)
{
  
    // Initialize with zero
    $number = 0;
    $count1 = 0;
  
    for ($i = 0; $i < $n; $i++) 
    {
  
        // Instead of converting all prefixes
        // to decimal, take reminder with x
        $number = ($number * 2 + $arr[$i]) % $x;
  
        // If number is divisible by x
        // then reminder = 0
        if ($number == 0)
            $count1 += 1;
    }
  
    return $count1;
}
  
// Driver code
$arr = array(1, 0, 1, 0, 1, 1, 0);
$n = sizeof($arr);
$x = 2;
echo CntDivbyX($arr, $n, $x);
  
// This code is contributed by Akanksha Rai
?>

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Output:

3

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



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