Question 59
Let $p$ be an odd prime number and let $(a_1, a_2, \dots, a_p)$ and $(b_1, b_2, \dots, b_p)$ be arbitrary arrangements of the $p$-tuple $(0, 1, \dots, p-1)$. For each $i$, let $c_i$ be the non-negative remainder when the product $a_i b_i$ is divided by $p$. Show that $(c_1, c_2, \dots, c_p)$ cannot be a rearrangement of $(0, 1, \dots, p-1)$.