# 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++

`/* 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; ` `} ` |

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## 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)); ` ` ` `} ` `} ` |

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## 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. ` |

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## 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 ` |

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## 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 ` `?> ` |

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Output:

4

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