<s>[INST] Find the set of $n\in\Bbb Z^+$ with $M=\{n,n+1,n+2,n+3,n+4,n+5\}$ partitionable into two sets 
Find the set of all positive integers $n$ with the property that the set $M=\{n, n + 1,n + 2,n + 3,n + 4,n + 5\}$ can be partitioned into two sets such that the product of the numbers in one set equals the product of the numbers in the other set.

If $n=1$ them $M=\{1,2,3,4,5,6\}$ and there is no such partition, so $n \ge 2$.
If the prime $p|n$ then either $p|2$ or $p|3$ or $p|5$ which means either $p=2$ or $p=3$ or $p=5$.
Suppose $n=2k$. Then $M=\{2k, 2k + 1,2k + 2,2k + 3,2k + 4,2k + 5\}$. I have no idea how to proceed.
 [/INST] Hint, given
$$n \equiv 0 \pmod{n}$$
$$n+1 \equiv 1 \pmod{n}$$
$$n+2 \equiv 2 \pmod{n}$$
$$n+3 \equiv 3 \pmod{n}$$
$$n+4 \equiv 4 \pmod{n}$$
$$n+5 \equiv 5 \pmod{n}$$
if there exists such a partition, $Q\cap S=\varnothing$ and $Q\cup S=\left \{ 0,1,2,3,4,5 \right \}$ 
$$\prod_{q \in Q} \left ( n+q \right ) = \prod_{s \in S} \left ( n+s \right ) \Rightarrow \prod_{q \in Q} \left ( n+q \right ) \equiv \prod_{s \in S} \left ( n+s \right ) \pmod{n}$$
which is the same as
$$\prod_{q \in Q} q \equiv \prod_{s \in S} s \pmod{n}$$
And $0$ will be on one side, let's suppose the right one
$$\prod_{q \in Q} q \equiv 0 \pmod{n}$$
which has no solutions for $n > 120$. This leaves us with just a handful of cases to analyse.
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