Problem 4

Find all positive integers

n

such that

\{n,n+1,n+2,n+3,n+4,n+5\}

can be partitioned into two disjoint sets whose products are equal.

Step 4 of 4: Reject the candidate

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In plain words

An equal-product partition forces every prime exponent in the total product to be even.

5\\text{ occurs to an odd power in }1\\cdot2\\cdot3\\cdot4\\cdot5\\cdot6

Detailed analysis

In $\{1,2,3,4,5,6\}$ , the prime $5$ divides exactly one element and occurs to the first power in the total product. An equal partition would make the total product a square, so this is impossible. Thus no positive $n$ works.