1. Does there exist a positive integer divisible by 2020, in whose representation the digits $0,1, \cdots, 9$ appear the same number of times?
1. Existence.

For example, the number 12123434565679798080 satisfies the requirement.

Since $2020=101 \times 20$, and $(101,20)=1$, it is only necessary to verify the divisibility of the number by 101 and 20 separately.
Obviously, the number is divisible by 20.
To verify that the number is divisible by 101, it is sufficient to point out that any positive integer of the form $a 0 a 00 \cdots 0$ is divisible by 101.

In fact, the number can be expressed as the sum of several such numbers.