It's hard to keep up with the fashionable trends of the day, and mathematical fashion is no exception. The fashion season of 2024 has the following unspoken fashion rules: 1 is fashionable. If a and b are fashionable numbers, then so is 1 a+b' All other numbers are "so last year" and thus not fashionable. (a) Find the smallest and largest fashionable numbers, and prove that they are the smallest and largest. (b) Which numbers are fashionable? Can you characterize the set of all fashionable numbers? (c) Word on the street is that in 2025, the fashion rules will be the same except for "1 is fashionable". Explore how the set of fashionable numbers changes if we begin with a different starting set of numbers. For example, what if 2 and 3 are the given fashionable numbers?