10.1. Kolya wrote a ten-digit number on the board, consisting of different digits. Sasha added one digit so that the resulting number would be divisible by 9. Which digit could Sasha have added?
Answer: 0 or 9.

Solution: The sum of all 10 digits from 0 to 9 is 45. Sasha needs to append a digit so that the sum of the digits in the resulting number is divisible by 9 (the divisibility rule for 9). The sum of the digits will be divisible by 9 if 0 or 9 is appended.

Criteria: Only 1 point for the correct answer 0 or $9-1$ point. 2 points for the correct answer 0 and 9.