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

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>` |

**Output: **

4

This article is contributed by **Utkarsh Trivedi**. If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.

Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.

Attention reader! Don’t stop learning now. Get hold of all the important mathematical concepts for competitive programming with the **Essential Maths for CP Course** at a student-friendly price. To complete your preparation from learning a language to DS Algo and many more, please refer **Complete Interview Preparation Course****.**