Question 38
Given any $n$ ($\gt 1$) coprime positive integers $a_1, a_2, \dots, a_n$, denote $A = a_1 + a_2 + \dots + a_n$. Let $d_i = (A, a_i)$ (the greatest common divisor), $i = 1, 2, \dots, n$.
Let $D_i$ be the greatest common divisor of $\{a_1, a_2, \dots, a_n\} \setminus \{a_i\}$, $i = 1, 2, \dots, n$. Find the minimum of $\prod_{i=1}^n \frac{A - a_i}{d_i D_i}$.
(posed by Zhang Sihui)
Let $D_i$ be the greatest common divisor of $\{a_1, a_2, \dots, a_n\} \setminus \{a_i\}$, $i = 1, 2, \dots, n$. Find the minimum of $\prod_{i=1}^n \frac{A - a_i}{d_i D_i}$.
(posed by Zhang Sihui)