Question 3
Let $n\ge3$ be an integer. Prove that there exist positive integers $a_1,a_2,\ldots,a_n$, each at least $2$, such that $$a_1a_2\cdots\widehat{a_i}\cdots a_n\equiv1\pmod{a_i},\qquad i=1,\ldots,n.$$ Here $\widehat{a_i}$ means the term $a_i$ is omitted.