, 2020

There are $2019$ penguins waddling towards their favourite restaurant. As the penguins arrive, they are handed tickets numbered in ascending order from $1$ to $2019$, and told to join the queue. The first penguin starts the queue. For each $n > 1$ the penguin holding ticket number $n$ finds the greatest $m < n$ which divides $n$ and enters the queue directly behind the penguin holding ticket number $m$. This continues until all $2019$ penguins are in the queue.

(a) How many penguins are in front of the penguin with ticket number $2$?

(b) What numbers are on the tickets held by the penguins just in front of and just behind the penguin holding ticket $33$?