Question 40
For any integer $n$ with $n \gt 1$, let
$$
D(n) = \{a - b \mid a, b \text{ are positive integers with } n = ab \text{ and } a \gt b\}.
$$
Prove that for any integer $k$ with $k \gt 1$, there exist $k$ pairwise distinct integers $n_1, n_2, \dots, n_k$ with $n_i \gt 1\ (1 \le i \le k)$, such that $D(n_1) \cap D(n_2) \cap \dots \cap D(n_k)$ has at least two elements.
$$
D(n) = \{a - b \mid a, b \text{ are positive integers with } n = ab \text{ and } a \gt b\}.
$$
Prove that for any integer $k$ with $k \gt 1$, there exist $k$ pairwise distinct integers $n_1, n_2, \dots, n_k$ with $n_i \gt 1\ (1 \le i \le k)$, such that $D(n_1) \cap D(n_2) \cap \dots \cap D(n_k)$ has at least two elements.