I am stuck in the following problem, could someone kindly give me some ideas? Thanks in advance.
This is the statement of the problem:
Prove that for any integer $n \ge 2,$ there exists a set of $n$ distinct integers such that for any $a,\,b$ in that set $(a-b)^2$ divides $ab$.
This is actually a problem from USAMO 1998. The solution can be found in the book "104 Number Theory Problems" from Titu Andreescu, Dorin Andrica and Zuming Feng.
The original solution is shown below. But I can prove a more strict version, limited to natural numbers only (the original solution includes zero). Suppose that $a>b$:
$$a-b=k,\quad ab=b(b+k)$$
$$(a-b)^2\mid ab \iff k^2\mid b^2+bk$$
This is always true if $k\mid b$.
In other words, if all pairs $(a,b)$ picked from the set satisfy the following criteria:
$$ (a-b)\mid b$$
...that set is a solution of the problem. Such set consisting of natural numbers only definitely exists for any given $n$ and a fairly simple proof can be found HERE
Just for reference, here is the solution from the book: