$3|A$ so $3|n$ and for $n=3k$ we can take $A=7077070770(\text{repeat k times})...70770$

The multiple of $2022$ contains only $7$ and $0$. So it must contain $1$ and $0$, before multiplying by $7$, so a quick trivial thought is to multiply $2022$ by $5$ and we get $10110$, so in this case $n=3$, Now multiply it by $(10^5 + 1)$ you with get the next such number with $n=6$ and so on. So all possible values of $n$ are multiples of $3$. Note that if you wish you can also multiply $10110$ by $(10^k + 1)$ where $k>5$ and $k$ is a natural number.