Suppose that $\displaystyle x \in \mathbb{Z_p^\times}$ has order $\displaystyle p-1$. Prove that the integers $\displaystyle x^1, x^2, ..., x^{p-1}$ are incongruent mod $\displaystyle p$.
Suppose that $\displaystyle x \in \mathbb{Z_p^\times}$ has order $\displaystyle p-1$. Prove that the integers $\displaystyle x^1, x^2, ..., x^{p-1}$ are incongruent mod $\displaystyle p$.
This is precisely what it means for $\displaystyle x$ to have order p-1.