8.1. A four-digit number is called "beautiful" if it is impossible to append a digit to the right so that the resulting five-digit number is divisible by 11. How many beautiful numbers are there that are greater than 3100 and less than 3600?
Answer: 46

Solution. Note that the number 3101 is beautiful because 31009 and 31020 are divisible by 11, so 3101* cannot be divisible by 11. Suppose the number is beautiful, then we have found 10 consecutive numbers, none of which are divisible by 11, but among 11 consecutive numbers, at least one is divisible by 11, which means the next number is divisible by 11. This number is one more than our four-digit number to which we appended 0. Therefore, the number $\overline{3 a b c}$ is beautiful if $\overline{3 a b c}+1$ is divisible by 11.

Note that the smallest number in the given set that is divisible by 11 is 3102, and the largest is 3597. There are as many such numbers as there are quotients from dividing them by $11: 282, \ldots, 327$, and this is $327-281=46$ numbers.