Biggest integer which has maximum digit sum in range from 1 to n

2.5

Given a number n, find a number in range from 1 to n such that its sum is maximum. If there are several such integers, determine the biggest of them.

Examples:

Input:  n = 100
Output: 99  
99 is the largest number in range from
1 to 100 with maximum sum of digits.

Input: n = 48
Output: 48
Explanation: 
There are two numbers with maximum
digit sum. The numbers are 48 and 39
Since 48 > 39, it is our answer.

A naive approach is to iterate for all numbers from 1 to n and find out which number has maximum sum of digits. Time complexity of this solution is O(n).

An efficient approach is to iterate from n to 1. Do the following for each digit of current number, if the digit is not zero, reduce it by one and change all other digits to nine to the right of it. If the sum of digits in the resulting integer is strictly greater than the sum of the digits of the current answer, then update the answer with the resulting integer. If the sum of the resulting integer is same as the current answer, then if the resulting integer is more then current answer, update the current answer with the resulting integer.

How to reduce a digit and change all other digits on its right to 9?
Let x be our current number. We can find next number for current digit using below formula. In below formula, b is a power of 10 to represent position of current digit. After every iteration we reduce x to x/10 and change b to b * 10.

We use (x – 1) * b + (b – 1);
This line can further be explained as, if the number is x = 521 and b = 1, then

  • (521 – 1) * 1 + (1-1) which gives you 520, which is the thing we need to do, reduce the position number by 1 and replace all other numbers to the right by 9.
  • After x /= 10 gives you x as 52 and b*=10 gives you b as 10, which is again executed as (52-1)*(10) + 9 which gives you 519, which is what we have to do, reduce the current index by 1 and increase all other rights by 9.
// CPP program to find the number with maximum digit
// sum.
#include <bits/stdc++.h>
using namespace std;

// function to calculate the sum of digits of a number.
int sumOfDigits(int a)
{
    int sum = 0;
    while (a) {
        sum += a % 10;
        a /= 10;
    }
    return sum;
}

// Returns the maximum number with maximum sum of digits.
int findMax(int x)
{
    // initializing b as 1 and initial max sum to be of n
    int b = 1, ans = x;

    // iterates from right to left in a digit
    while (x) {

        // while iterating this is the number 
        // from right to left
        int cur = (x - 1) * b + (b - 1);

        // calls the function to check if sum of
        // cur is more then of ans
        if (sumOfDigits(cur) > sumOfDigits(ans) || 
           (sumOfDigits(cur) == sumOfDigits(ans) && 
            cur > ans))
            ans = cur;

        // reduces the number to one unit less
        x /= 10;
        b *= 10;
    }

    return ans;
}

// driver program to test the above function
int main()
{
    int n = 521; 
    cout << findMax(n); 
    return 0;
}

Output:

499

Time complexity: O(m) where m is the number of digits in n.

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