2. (20 points) How many even 100-digit numbers exist, each digit of whose decimal representation is one of the digits 0, 1, or 3?
Answer: $2 \cdot 3^{98}$.

Solution: According to the condition, the number must be even, so the least significant digit can only be the digit 0; for the hundredth
place, we have two options - the digit 1 or the digit 3; for the other places (from the second to the ninety-ninth) - three options - any of the digits $1,2,3$. Therefore, the total number of $100-$ digit numbers that satisfy the conditions will be $2 \cdot 3^{98} \cdot 1$.