Question 84
Find the number of tuples $(a_1, a_2, \dots, a_{2100})$ of integers between $0$ and $2099$ inclusive that satisfy the following condition:
There exists a tuple of integers $(b_1, b_2, \dots, b_{2100})$ such that for every integer $i$ from $1$ to $2100$,
$$
a_i \equiv \sum_{\substack{\gcd(j-i, 2100)=1 \\ 1 \le j \le 2100}} b_j \pmod{2100}
$$
holds.
There exists a tuple of integers $(b_1, b_2, \dots, b_{2100})$ such that for every integer $i$ from $1$ to $2100$,
$$
a_i \equiv \sum_{\substack{\gcd(j-i, 2100)=1 \\ 1 \le j \le 2100}} b_j \pmod{2100}
$$
holds.