KöMaL 2018 A728

Floyd the flea makes jumps on the positive integers. On the first day he can jump to any positive integer. From then on, every day he jumps to another number that is not more than twice his previous day's place. a) Show that Floyd can make infinitely many jumps in such a way that he never arrives at any number with the same sum of decimal digits as at a previous place. b) Can the flea jump this way if we consider the sum of binary digits instead of decimal digits?