Problem

Source: 2022 Czech-Polish-Slovak Match Junior, team p2 CPSJ

Tags: number theory, divisible



The number $2022$ is written on the board. In each step, we replace one of the $2$ digits with the number $2022$. For example $$2022 \Rightarrow 2020222 \Rightarrow 2020220222 \Rightarrow ...$$After how many steps can a number divisible by $22$ be written on the board? Specify all options.