Question

Let there be a sequence of integers a_1, a_2, a_3,.... For each a_i, multiply a_i written in base 10 by 5^100. Then, in the product, replace each digit with its remainder when divided by 2. Read this new number as if it’s in base 2, and convert it into base 10. This is a_(i+1). Prove that if a_1 is a positive integer less than 2^101, this sequence is periodic and has period 2^k for some k.


I experimented with lower exponents (2, 3, 4) since 100 is a large number. However, I couldn’t find anything that will help me. Any help please?