Question 164
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 $g: \mathbb{Z}/n\mathbb{Z} \to \mathbb{Z}/n\mathbb{Z}$, such that the 101 functions
$$
g(x),\ g(x) + x,\ g(x) + 2x,\ \dots,\ g(x) + 100x
$$
are all bijections on $\mathbb{Z}/n\mathbb{Z}$.
$$
g(x),\ g(x) + x,\ g(x) + 2x,\ \dots,\ g(x) + 100x
$$
are all bijections on $\mathbb{Z}/n\mathbb{Z}$.