One hundred brownies (girl scouts) are sitting in a big circle around the camp fire. Each brownie has one or more chestnuts and no two brownies have the same number of chestnuts. Each brownie divides her number of chestnuts by the number of chestnuts of her right neighbour and writes down the remainder on a green piece of paper. Each brownie also divides her number by the number of chestnuts of her left neighbour and writes down the remainder on a red piece of paper. For example, if Anja has $23$ chestnuts and her right neighbour Bregje has $5$, then Anja writes $3$ on her green piece of paper and Bregje writes $5$ on her red piece of paper.

If the number of distinct remainders on the $100$ green pieces of paper equals $2$, what is the smallest possible number of distinct remainders on the $100$ red pieces of paper?