Konstantin has a pile of $100$ pebbles. In each move, he chooses a pile and splits it into two smaller ones until he gets $100 $ piles each with a single pebble. (a) Prove that at some point, there are $30$ piles containing a total of exactly $60$ pebbles. (b) Prove that at some point, there are $20$ piles containing a total of exactly $60$ pebbles. (c) Prove that Konstantin may proceed in such a way that at no point, there are $19$ piles containing a total of exactly $60$ pebbles.