The remainder when $(2021)^{2023}$ is divided by $7$ is

Among the statements:
$(S1):$ $2023^{2022} - 1999^{2022}$ is divisible by $8$.
$(S2):$ $13(13^{n}) - 11n - 13$ is divisible by $144$ for infinitely many $n \in N$.

If the remainder when $x$ is divided by $4$ is $3$,then the remainder when $(2020+x)^{2022}$ is divided by $8$ is ....... .

The remainder left when $8^{2n} - 62^{2n+1}$ is divided by $9$ is:

For all integers $n \geq 1$,which of the following is divisible by $9$?

The remainder when $10^{10} \cdot (10^{10} + 1) \cdot (10^{10} + 2)$ is divided by $6$ is