Here’s the problem that I’m working on:
Let there be an infinite sequence of positive integers $a_1, a_2, a_3,...$. For each $a_i,$ multiply $a_i$ in base 10 by $5^{100}$, replace each digit with its remainder when divided by 2, and read the new number in binary. Call this number $a_{i+1}.$ If $a_1$ is any positive integer, show that for all large enough k, $a_k=a_{k+2^{100}}.$
By considering the number of digits, I realized that if $a_i$ is larger than $2^{101}$, the sequence eventually decreases until a term is less than $2^{101},$ so we just have to consider when $a_1$ is less than $2^{101}$. Further, I noticed that there’s a bijection between the numbers less than $2^{100}$ and the numbers between $2^{100}$ and $2^{101},$ so that might help. I also tried some sort of induction argument, but that got nowhere. I experimented with values smaller than 100, like $5^2, 5^3,$ etc and found that they all satisfy the problem conditions. However, I couldn’t find a pattern or anything useful, so I’m stuck here. Thanks for the help.