Question 128
Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1, a_2, \dots, a_n$, such that $a_1 a_2 \cdots \hat{a}_i \cdots a_n \equiv 1 \pmod{a_i}$, for $i = 1, \dots, n$. Here $\hat{a}_i$ means the term $a_i$ is omitted.