Question 156
Let $n$ be an odd natural number, and let $a_{1}, \ldots, a_{n}$ be nonzero natural numbers. Let $A$ denote the product of the integers $a_{i}$, and $d$ their greatest common divisor.
Show that
$$
\operatorname{GCD}\left(a_{1}^{n}+A, a_{2}^{n}+A, \ldots, a_{n}^{n}+A\right) \leqslant 2 d^{n}
$$
Show that
$$
\operatorname{GCD}\left(a_{1}^{n}+A, a_{2}^{n}+A, \ldots, a_{n}^{n}+A\right) \leqslant 2 d^{n}
$$