Count numbers with difference between number and its digit sum greater than specific value

Last Updated : 22 Apr, 2021

Given a positive value N, we need to find the count of numbers smaller than N such that the difference between the number and sum of its digits is greater than or equal to given specific value diff.
Examples:

Input : N = 13, diff = 2
Output : 4
Then 10, 11, 12 and 13 satisfy the given
condition shown below,
10 – sumofdigit(10) = 9 >= 2
11 – sumofdigit(11) = 9 >= 2
12 – sumofdigit(12) = 9 >= 2
13 – sumofdigit(13) = 9 >= 2  
Whereas no number from 1 to 9 satisfies 
above equation so final result will be 4

We can solve this problem by observing a fact that for a number k less than N,

 
if k – sumofdigit(k) >= diff then
above equation will be true for (k+1)
also because we know that sumofdigit(k+1)
is not greater than sumofdigit(k) + 1
so, k + 1 - sumofdigit(k + 1) >= 
k - sumofdigit(k)
but we know that right side of above 
inequality is greater than diff, 
so left side will also be greater than 
diff.

So, finally we can say that if a number k satisfies the difference condition then (k + 1) will also satisfy same equation so our job is to find the smallest number which satisfies the difference condition then all numbers greater than this and up to N will satisfy the condition so our answer will be N – smallest number we found.
We can find the smallest number satisfying this condition using binary search so total time complexity of solution will be O(log N)

C++

/* C++ program to count total numbers which
have difference with sum of digits greater
than specific value */
#include <bits/stdc++.h>
using namespace std;
 
// Utility method to get sum of digits of K
int sumOfDigit(int K)
{
    // loop until K is not zero
    int sod = 0;
    while (K)
    {
        sod += K % 10;
        K /= 10;
    }
    return sod;
}
 
// method returns count of numbers smaller than N,
// satisfying difference condition
int totalNumbersWithSpecificDifference(int N, int diff)
{
    int low = 1, high = N;
 
    // binary search while loop    
    while (low <= high)
    {
        int mid = (low + high) / 2;
 
        /* if difference between number and its sum
        of digit is smaller than given difference
        then smallest number will be on left side */
        if (mid - sumOfDigit(mid) < diff)        
            low = mid + 1;
         
        /* if difference between number and its sum
        of digit is greater than or equal to given
        difference then smallest number will be on
        right side */
        else       
            high = mid - 1;        
    }
 
    // return the difference between 'smallest number
    // found' and 'N' as result
    return (N - high);
}
 
// Driver code to test above methods
int main()
{
    int N = 13;
    int diff = 2;
 
    cout << totalNumbersWithSpecificDifference(N, diff);    
    return 0;
}

Java

/*  Java program to count total numbers which
   have difference with sum of digits greater 
   than specific value */
 
class Test
{
    //  Utility method to get sum of digits of K
    static int sumOfDigit(int K) 
    {
        //  loop until K is not zero
        int sod = 0;
        while (K != 0)
        {
            sod += K % 10;
            K /= 10;
        }
        return sod;
    }
     
    // method returns count of numbers smaller than N, 
    // satisfying difference condition
    static int totalNumbersWithSpecificDifference(int N, int diff)
    {
        int low = 1, high = N;
      
        //  binary search while loop    
        while (low <= high) 
        {
            int mid = (low + high) / 2;
      
            /* if difference between number and its sum 
               of digit is smaller than given difference 
               then  smallest number will be on left side */
            if (mid - sumOfDigit(mid) < diff)        
                low = mid + 1;
              
            /* if difference between number and its sum 
               of digit is greater than or equal to given 
               difference then  smallest number will be on 
               right side */
            else      
                high = mid - 1;        
        }
      
        // return the difference between 'smallest number 
        // found' and 'N' as result
        return (N - high);
    }
 
    // Driver method
    public static void main(String args[])
    {
        int N = 13;
        int diff = 2;
      
        System.out.println(totalNumbersWithSpecificDifference(N, diff)); 
    }
}

Python3

# Python program to count total numbers which
# have difference with sum of digits greater
# than specific value
 
 
# Utility method to get sum of digits of K
def sumOfDigit(K):
 
    # loop until K is not zero
    sod = 0
    while (K):
     
        sod =sod + K % 10
        K =K // 10
     
    return sod
 
 
# method returns count of
# numbers smaller than N,
# satisfying difference condition
def totalNumbersWithSpecificDifference(N,diff):
 
    low = 1
    high = N
 
    # binary search while loop    
    while (low <= high):
     
        mid = (low + high) // 2
 
        ''' if difference between number and its sum
        of digit is smaller than given difference
        then smallest number will be on left side'''
        if (mid - sumOfDigit(mid) < diff):        
            low = mid + 1
 
        # if difference between number and its sum
        # of digit greater than equal to given
        # difference then smallest number will be on
        # right side    
        else:
             
            high = mid - 1       
     
 
    # return the difference between 'smallest number
    # found' and 'N' as result
    return (N - high)
 
# Driver code to test above methods
N = 13
diff = 2
 
print(totalNumbersWithSpecificDifference(N, diff))    
     
# This code is contributed by Anant Agarwal.

C#

// C# program to count total numbers 
// which have difference with sum of  
// digits greater than specific value 
using System;
 
class Test {
     
    // Utility method to get sum
    // of digits of K
    static int sumOfDigit(int K) 
    {
         
        // loop until K is not zero
        int sod = 0;
        while (K != 0)
        {
            sod += K % 10;
            K /= 10;
        }
        return sod;
    }
     
    // method returns count of numbers 
    // smaller than N, satisfying 
    // difference condition
    static int totalNumbersWithSpecificDifference(int N,
                                                  int diff)
    {
        int low = 1, high = N;
     
        // binary search while loop 
        while (low <= high) 
        {
            int mid = (low + high) / 2;
     
            // if difference between number and
            // its sum of digit is smaller than 
            // given difference then smallest
            // number will be on left side 
            if (mid - sumOfDigit(mid) < diff)     
                low = mid + 1;
             
            // if difference between number and  
            // its sum of digit is greater than  
            // or equal to given difference then  
            // smallest number will be on right side 
            else   
                high = mid - 1;     
        }
     
        // return the difference between 
        // 'smallest number found' 
        // and 'N' as result
        return (N - high);
    }
 
    // Driver code
    public static void Main()
    {
        int N = 13;
        int diff = 2;
     
        Console.Write(totalNumbersWithSpecificDifference(N, diff)); 
    }
}
 
// This code is contributed by nitin mittal

PHP

<?php
// PHP program to count total numbers which
// have difference with sum of digits greater 
// than specific value
 
// method to get sum of digits of K
function sumOfDigit($K) 
{
     
    // loop until K is not zero
    $sod = 0;
    while ($K)
    {
        $sod += $K % 10;
        $K /= 10;
    }
    return $sod;
}
 
// method returns count of 
// numbers smaller than N, 
// satisfying difference condition
function totalNumbersWithSpecificDifference($N, $diff)
{
    $low = 1; $high = $N;
 
    // binary search while loop 
    while ($low <= $high) 
    {
        $mid = floor(($low + $high) / 2);
 
        /* if difference between number and its sum 
           of digit is smaller than given difference 
           then smallest number will be on left side */
        if ($mid - sumOfDigit($mid) < $diff)     
            $low = $mid + 1;
         
        /* if difference between number and its sum 
           of digit is greater than or equal to given 
           difference then smallest number will be on 
           right side */
        else   
            $high = $mid - 1;     
    }
 
    // return the difference 
    // between 'smallest number 
    // found' and 'N' as result
    return ($N - $high);
}
 
// Driver Code
$N = 13;
$diff = 2;
echo totalNumbersWithSpecificDifference($N, $diff); 
 
// This code is contributed by nitin mittal
?>

Javascript

<script>
 
// javascript program to
// count total numbers which
// have difference with sum
// of digits greater
// than specific value
 
// method to get sum of digits of K
function sumOfDigit(K)
{
     
    // loop until K is not zero
    let sod = 0;
    while (K)
    {
        sod += K % 10;
        K /= 10;
    }
    return sod;
}
 
// method returns count of
// numbers smaller than N,
// satisfying difference condition
function
totalNumbersWithSpecificDifference(N, diff)
{
    let low = 1; 
    let high = N;
 
    // binary search while loop
    while (low <= high)
    {
        let mid = Math.floor((low + high) / 2);
 
        /* if difference between number and its sum
        of digit is smaller than given difference
        then smallest number will be on left side */
        if (mid - sumOfDigit(mid) < diff)    
            low = mid + 1;
         
        /* if difference between number and its sum
        of digit is greater than or equal to given
        difference then smallest number will be on
        right side */
        else   
            high = mid - 1;    
    }
 
    // return the difference
    // between 'smallest number
    // found' and 'N' as result
    return (N - high);
}
 
// Driver Code
let N = 13;
let diff = 2;
document.write( 
totalNumbersWithSpecificDifference(N, diff)
);
 
// This code is contributed by Bobby
 
</script>