KMO 2021 Problem 3

Show that for any positive integers $k$ and $1 \leq a \leq 9$ , there exists $n$ such that satisfies the below statement.

When $2^n=a_0+10a_1+10^2a_2+ \cdots +10^ia_i+ \cdots$ $(0 \leq a_i \leq 9$ and $a_i$ is integer), $a_k$ is equal to $a$ .