10.1. Append a digit to the left and right of the eight-digit number 20222023 so that the resulting 10-digit number is divisible by 72. (List all possible solutions.)
Answer: 3202220232.

Solution. Since $72=8 \cdot 9$, it is required to append digits so that the resulting number is divisible by both 8 and 9. Divisibility by 8 is determined by the last three digits: thus, to the two-digit number 23, we need to append a digit on the right to form a three-digit number that is a multiple of 8. There is only one digit, namely 2, for which this is possible ( $232: 8=29$ ). After this, the first digit of the desired number is determined by the divisibility rule for 9 (the sum of the digits must be divisible by 9, so we need to add 3 to the existing sum of digits, which is 15).