9.1. To the eight-digit number 20192020, append a digit on the left and a digit on the right so that the resulting 10-digit number is divisible by 72. List all possible solutions.
Answer: 2201920200 or 3201920208. 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: to twenty, we need to append a digit on the right so that a three-digit number divisible by 8 is obtained. This can be done in two ways: we get 200 or 208. 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). Thus, this digit will be, respectively, 2 or 3.