Question 89
Let $p$ be a prime number. Prove that there exists a permutation $\left(a_{1}, \ldots, a_{p}\right)$ of $(1, \ldots, p)$ such that the integers $a_{1}, a_{1} \cdot a_{2}, \ldots, a_{1} \cdot a_{2} \cdot \ldots \cdot a_{p}$ leave $p$ pairwise distinct remainders upon Euclidean division by $p$.