US-TST 2019 Problem 2

Let $\mathbb{Z} / n \mathbb{Z}$ denote the set of integers considered modulo $n$ (hence $\mathbb{Z} / n \mathbb{Z}$ has $n$ elements). Find all positive integers $n$ for which there exists a bijective function $g: \mathbb{Z} / n \mathbb{Z} \rightarrow \mathbb{Z} / n \mathbb{Z}$, such that the 101 functions $$g(x), g(x)+x, g(x)+2 x, \ldots, g(x)+100 x$$ are all bijections on $\mathbb{Z} / n \mathbb{Z}$.