This question was sent to me by my friend.
Let there be a sequence of positive integers $a_1, a_2, \dots$. For all $a_i$, multiply $a_i$ in base $10$ by $5^{2020}$, replace each digit with its remainder when divided by 2, and read the result as a binary number. Call this new number $a_{i+1}$. If $a_1$ is any positive integer, show that $a_k=a_{k+2^{2020}}$ for large enough $k$.
I managed to show that any number greater than $2^{2021}$ will eventually reduce to a number less than $2^{2021}$, so we just have to consider numbers less than $2^{2021}$. I thought of using pigeonhole principle, but that didn't help in showing that the period would be a divisor of $2^{2020}$. I tried induction and experimenting with smaller exponents, but I couldn't make any progress.
Thanks in advance for the help.