A number theory problem by Pankaj Kumar Yadav

It is straightforward that the solution cannot be a three digit or larger number, as the maximum sum of digits for an n-digit number is 9n, which is less than 10^(n-1) for n>2.

A two digit number xy is represented as 10x+y which must be equal to x+y. Solving for x yields a result of zero, meaning that the answer can only have one digit, and 9 is the largest.

it is easy to see that the number of digits of the number is less than three,as the minimum number of three digits is 100 while the max sum of digits is 9+9+9=27.Using similar reasoning, it cannot be of 4 digits,5 digits... No number of two digits greater than 18 has this property as the maximum possible sum of digits of a 2 digit number is 9+9=18.no number out of 10,11,12...18 has the sum of digits equals to itself, as the maximum possible sum is 1+8=9<10. every one digit number satisfies this. Largest 1 digit number is 9, which is the answer.