Question 59
Let $n$ be an odd positive integer with $n\gt 1$ and let $a_{1}, a_{2}, \ldots, a_{n}$ be positive integers such that $\operatorname{gcd}\left(a_{1}, a_{2}, \ldots, a_{n}\right)=1$. Let $d=\operatorname{gcd}\left(a_{1}^{n}+a_{1} a_{2} \cdots a_{n}, a_{2}^{n}+a_{1} \cdot a_{2} \cdots a_{n}, \ldots, a_{n}^{n}+a_{1} \cdot a_{2} \cdots a_{n}\right)$. Show that the possible values of $d$ are $d=1, d=2$.