Sub-string Divisibility by 3 Queries

0 0 5 minutes read

Given a large number, n (having number digits up to 10^6) and various queries of the form :
Query(l, r) : find if the sub-string between the indices l and r (Both inclusive) are divisible by 3.
Examples:

Input: n = 12468236544
Queries:
l=0 r=1
l=1 r=2
l=3 r=6
l=0 r=10
Output:
Divisible by 3 
Divisible by 3 
Not divisible by 3
Divisible by 3

Explanation:
In the first query, 12 is divisible by 3
In the second query, 24 is divisible by 3 and so on.

We know that any number is divisible by 3 if the sum of its digits is divisible by 3. Hence the idea is to pre-process an auxiliary array that would store the sum of digits.

Mathematically,
sum[0] = 0
and 
for i from 0 to number of digits of number:
    sum[i+1] = sum[i]+ toInt(n[i])
where toInt(n[i]) represents the integer value 
of i'th digit of n

Once our auxiliary array is processed, we can answer each query in O(1) time, because the substring from indices l to r would be divisible by 3 only if, (sum[r+1]-sum[l])%3 == 0
Below is a the implementation program for the same.

C++

// C++ program to answer multiple queries of
// divisibility by 3 in substrings of a number
#include <iostream>
using namespace std;
 
// Array to store the sum of digits
int sum[1000005];
 
// Utility function to evaluate a character's
// integer value
int toInt(char x)
{
    return int(x) - '0';
}
 
// This function receives the string representation
// of the number and precomputes the sum array
void prepareSum(string s)
{
    sum[0] = 0;
    for (int i=0; i<s.length(); i++)
        sum[i+1] = sum[i] + toInt(s[i]);
}
 
// This function receives l and r representing
// the indices and prints the required output
void query(int l,int r)
{
    if ((sum[r+1]-sum[l])%3 == 0)
        cout << "Divisible by 3\n";
    else
        cout << "Not divisible by 3\n";
}
 
// Driver function to check the program
int main()
{
    string n = "12468236544";
 
    prepareSum(n);
    query(0, 1);
    query(1, 2);
    query(3, 6);
    query(0, 10);
    return 0;
}

Java

// Java program to answer multiple queries of
// divisibility by 3 in substrings of a number
class GFG 
{
 
    // Array to store the sum of digits
    static int sum[] = new int[1000005];
 
    // Utility function to evaluate a character's
    // integer value
    static int toInt(char x) 
    {
        return x - '0';
    }
 
    // This function receives the string representation
    // of the number and precomputes the sum array
    static void prepareSum(String s)
    {
        sum[0] = 0;
        for (int i = 0; i < s.length(); i++) 
        {
            sum[i + 1] = sum[i] + toInt(s.charAt(i));
        }
    }
 
    // This function receives l and r representing
    // the indices and prints the required output
    static void query(int l, int r) 
    {
        if ((sum[r + 1] - sum[l]) % 3 == 0) 
        {
            System.out.println("Divisible by 3");
        } 
        else
        {
            System.out.println("Not divisible by 3");
        }
    }
 
    // Driver code
    public static void main(String[] args)
    {
        String n = "12468236544";
 
        prepareSum(n);
        query(0, 1);
        query(1, 2);
        query(3, 6);
        query(0, 10);
    }
}
 
// This code has been contributed by 29AjayKumar

Python3

# Python3 program to answer multiple queries of
# divisibility by 3 in substrings of a number
 
 
# Array to store the sum of digits
sum = [0 for i in range(1000005)]
 
# Utility function to evaluate a character's
# integer value
def toInt(x):
 
    return int(x)
 
# This function receives the string representation
# of the number and precomputes the sum array
def prepareSum(s):
 
    sum[0] = 0
    for i in range(0, len(s)):
        sum[i + 1] = sum[i] + toInt(s[i])
 
# This function receives l and r representing
# the indices and prints the required output
def query(l, r):
 
    if ((sum[r + 1] - sum[l]) % 3 == 0):
        print("Divisible by 3")
    else:
        print("Not divisible by 3")
 
# Driver function to check the program
if __name__=='__main__':
     
    n = "12468236544"
    prepareSum(n)
    query(0, 1)
    query(1, 2)
    query(3, 6)
    query(0, 10)
 
# This code is contributed by
# Sanjit_Prasad

C#

// C# program to answer multiple queries of
// divisibility by 3 in substrings of a number
using System;
 
class GFG 
{
 
    // Array to store the sum of digits
    static int []sum = new int[1000005];
 
    // Utility function to evaluate a character's
    // integer value
    static int toInt(char x) 
    {
        return x - '0';
    }
 
    // This function receives the string representation
    // of the number and precomputes the sum array
    static void prepareSum(String s)
    {
        sum[0] = 0;
        for (int i = 0; i < s.Length; i++) 
        {
            sum[i + 1] = sum[i] + toInt(s[i]);
        }
    }
 
    // This function receives l and r representing
    // the indices and prints the required output
    static void query(int l, int r) 
    {
        if ((sum[r + 1] - sum[l]) % 3 == 0) 
        {
            Console.WriteLine("Divisible by 3");
        } 
        else
        {
            Console.WriteLine("Not divisible by 3");
        }
    }
 
    // Driver code
    public static void Main()
    {
        String n = "12468236544";
 
        prepareSum(n);
        query(0, 1);
        query(1, 2);
        query(3, 6);
        query(0, 10);
    }
}
 
/* This code contributed by PrinciRaj1992 */

Javascript

<script>
 
// JavaScript program to answer multiple queries of
// divisibility by 3 in substrings of a number
 
    // Array to store the sum of digits
    let sum = [];
   
    // Utility function to evaluate a character's
    // integer value
    function toInt(x) 
    {
        return x - '0';
    }
   
    // This function receives the string representation
    // of the number and precomputes the sum array
    function prepareSum(s)
    {
        sum[0] = 0;
        for (let i = 0; i < s.length; i++) 
        {
            sum[i + 1] = sum[i] + toInt(s[i]);
        }
    }
   
    // This function receives l and r representing
    // the indices and prints the required output
    function query(l, r) 
    {
        if ((sum[r + 1] - sum[l]) % 3 == 0) 
        {
            document.write("Divisible by 3" + "<br />");
        } 
        else
        {
            document.write("Not divisible by 3" + "<br />");
        }
    }
 
// Driver Code
 
        let n = "12468236544";
   
        prepareSum(n);
        query(0, 1);
        query(1, 2);
        query(3, 6);
        query(0, 10);
                         
</script>

PHP

<?php
     
  // PHP program to answer multiple queries of
// divisibility by 3 in substrings of a number
 
    // Array to store the sum of digits
    $sum = [];
   
    // Utility function to evaluate a character's
    // integer value
    function toInt($x) 
    {
        return $x - '0';
    }
   
    // This function receives the string representation
    // of the number and precomputes the sum array
    function prepareSum($s)
    {
        $sum[0] = 0;
        for ($i = 0; $i < strlen($s); $i++) 
        {
            $sum[$i + 1] = $sum[$i] + toInt($s[$i]);
        }
    }
   
    // This function receives l and r representing
    // the indices and prints the required output
    function query($l, $r) 
    {
       
        if (($sum[$r + 1] - $sum[$l]) % 3 == 0) 
        {
            echo("Divisible by 3");
        } 
        else
        {
            echo("Not divisible by 3");
        }
    }
 
// Driver Code
 
        $n = "12468236544";
   
        prepareSum($n);
        query(0, 1);
        query(1, 2);
        query(3, 6);
        query(0, 10);
   
  // This code is contributed by laxmigangarajula03
?>

Output:

Divisible by 3
Divisible by 3
Not divisible by 3
Divisible by 3

Time Complexity: O(n)

Auxiliary Space: O(n)
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