Let's call a natural number "remarkable" if it is the smallest among all natural numbers with the same digit sum as it has.

How many three-digit remarkable numbers exist?

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"Remarkable" numbers with the sum of digits from 1 to 18 are either single-digit or two-digit. Therefore, remarkable three-digit numbers have a sum of digits from 19 to 27. Each such sum corresponds to one remarkable number. Consequently, there are nine three-digit "remarkable" numbers.

## Answer

9 numbers.