Find the number which when added to any other number does not change the sum of its digit to ultimately single digit.

Any number of 9s will do the trick, so 99 (+100-1), 999 (+1000-1) etc will all work 198 will also work (200-2), as will 297 (300-3) and any "extension" (400-4, 500-5 etc, and 3000-3, 30000-3, etc). In short, the answer is any number of the form x*10^y-x. nimesh is also correct because y=0. x doesn't have to be less than 10; 125000-125 will also work for example (as will 1125: try it!).