SCHWEITZER 1949 Problem 3

Let $p$ be an odd prime number and $a_1,a_2,...,a_p$ and $b_1,b_2,...,b_p$ two arbitrary permutations of the numbers $1,2,...,p$ . Show that the least positive residues modulo $p$ of the numbers $a_1b_1, a_2b_2,...,a_pb_p$ never form a permutation of the numbers $1,2,...,p$ .