Numbers having difference with digit sum more than s
You are given two positive integer value n and s. You have to find the total number of such integer from 1 to n such that the difference of integer and its digit sum is greater than given s.
Input : n = 20, s = 5 Output :11 Explanation : Integer from 1 to 9 have diff(integer - digitSum) = 0 but for 10 to 20 they have diff(value - digitSum) > 5 Input : n = 20, s = 20 Output : 0 Explanation : Integer from 1 to 20 have diff (integer - digitSum) > 5
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The very first and basic approach to solve this question is to check for all integer starting from 1 to n and for each check whether integer minus digit sum is greater than s or not. This will become very time costly because we have to traverse 1 to n and for each integer we also have to calculate the digit sum.
Before moving to better approach lets have some key analysis about this questions and its features:
- For the largest possible integer (say long long int i.e. 10^18), the maximum possible digit sum is 9*18 (when all of digits are nine) = 162. This means in any case all the integer greater than s + 162 satisfy the condition of integer – digitSum > s.
- All integer less than s can not satisfy the given condition for sure.
- All the integers within a tens range (0-9, 10-19…100-109) does have same value of integer minus digitSum.
Using above three key features we can shorten our approach and time complexity in a manner where we have to iterate only over s to s+163 integers. Beside checking for all integer within range we only check for each 10th integer (e.g 150, 160, 170..).
// if n < s then return 0 if n<s return 0 else // iterate for s to min(n, s+163) for i=s to i min(n, s+163) // return n-i+1 if (i-digitSum)>s return (n-i+1) // if no such integer found return 0 return 0