Finding all positive integers $n$ such that $\{n,n+1,n+2,n+3,n+4,n+5\}$ partitions into two subsets such that the products of elements are the same

Question

Question from 12th IMO:

Find all positive integers $n\in\mathbb Z^+$ such that the set $\{n, n + 1, n + 2, n + 3, n + 4, n + 5\}$ can be split into two disjoint subsets such that the products of elements in these subsets are the same.

It seems simple to me, yet I can't find the solution. Edit: I have found no such examples.

Answer 1

We can see that $n+2$ divides the product of the rest of the numbers, and hence divides $(-2)(-1)(1)(2)(3)=12$. Similarly, $n+3$ divides $-12$, and hence, divides $12$. Both can hold simultaneously only if $n=1$. We can easily see that this fails (since only one of the values is divisible by $5$). Thus, there are no solutions in positive integers.