Let there be a sequence of integers a_1, a_2, a_3, ... For all a_i written as a base 10 number, multiply it by 5^100 and write it in base 10, replace each digit with its remainder when divided by 2, and read that number as if it's binary. Call that number a_(i+1). Prove that if a_1 is any positive integer less than 2^101, the sequence is periodic and has period 2^k for some k.