Find all such positive integers n that the set $(n,n+1,n+2,n+3,n+4,n+5)$ can be partitioned into two subsets such that product of the two subsets is equal.
closed as off-topic by S.C.B., JKnecht, Pragabhava, Daniel W. Farlow, Ben Sheller Apr 12 '16 at 18:43
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If $p>=5$ is a prime number, then $p$ cannot divide $n+1$, $n+2$, $n+3$ or $n+4$ (if it did, it would divide EXACTLY ONE of the six numbers, and you couldn't break them into two subsets with equal products). So the four numbers in the middle are divisible, at most, by 2 and 3. Two of them are odd, so they must be powers of 3, and the difference between them is 2. So either $n+1$, $n+3$ or $n+2$, $n+4$ must be 1 and 3. Which is not possible. So there are no solutions.