1113*. The number

$$
1010101 \ldots 0101
$$

is divisible by 9999. How many ones can be in the notation of this number? List all cases.
$\triangle$ Let's factorize the number 9999 into prime factors:

$$
9999=3^{2} \cdot 11 \cdot 101
$$

According to the divisibility rules for 9 and 11, the number of ones in the original number's representation must be divisible by 9 and 11.

A number is divisible by 101 if and only if, disregarding the first three digits in its representation, the sequence 0101 repeats, meaning the total number of ones is even. Therefore, the number of ones must be divisible by the product $2 \cdot 99=198$.

Answer: $198 k(k \in N)$.