Proving that there exist products of $a_k \equiv 1 \pmod {a_i}$

Question

Let $n>2$ be an integer. Prove that there exist numbers $a_1, a_2, \ldots ,a_n$ such that $$a_1a_2\cdots \widehat{a_i}\cdots a_n \equiv 1 \pmod{a_i}$$ for $i=1,2,3,\ldots,n$. Here $\widehat{a_i}$ means that $a_i$ is omitted.

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