8.1. A hundred-digit natural number $N$ is composed of ones and twos, and between any two twos, there is an even number of digits. It is known that $N$ is divisible by 3. How many ones and how many twos are in the number $N$?
Answer: two twos and 98 ones. Solution. If the number $N$ contains more than two twos, then considering any three twos, we get a contradiction: indeed, between the first and second twos there is an even number of digits, between the second and third twos there is an even number, but together with the second (i.e., the middle) two, it results in an odd number. On the other hand, the number $N$ must contain at least two twos, because otherwise the sum of the digits would be 100 (if there are no twos) or 101 (if there is one two), and $N$ would not be divisible by 3. In the case of two twos, all conditions are satisfied when there is an even number of ones between them.