Question 136
If $n$ is an integer, $n \ge 3$, and $a_1, a_2, \dots, a_n$ are non-zero integers such that
$$
a_1 a_2 \cdots a_n \left( \frac{1}{a_1^2} + \frac{1}{a_2^2} + \cdots + \frac{1}{a_n^2} \right)
$$
is an integer, does it follow that the product $a_1 a_2 \cdots a_n$ is divisible by each $a_i^2$?
$$
a_1 a_2 \cdots a_n \left( \frac{1}{a_1^2} + \frac{1}{a_2^2} + \cdots + \frac{1}{a_n^2} \right)
$$
is an integer, does it follow that the product $a_1 a_2 \cdots a_n$ is divisible by each $a_i^2$?