Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1,a_2,..,a_n$, such that $a_1 a_2 ... \widehat{a_i}... a_n \equiv 1$ (mod $a_i$), for $i = 1,..., n$. Here $\widehat{a_i}$ means the term $a_i$ is omitted.
Source: Singapore Senior Math Olympiad 2015 2nd Round p3 SMO
Tags: number theory, remainder
Let $n \ge 3$ be an integer. Prove that there exist positive integers $\ge 2$, $a_1,a_2,..,a_n$, such that $a_1 a_2 ... \widehat{a_i}... a_n \equiv 1$ (mod $a_i$), for $i = 1,..., n$. Here $\widehat{a_i}$ means the term $a_i$ is omitted.