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 2 of 4: Find the forced odd term

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In plain words

The three odd positions must distribute the only possible odd prime factors, leaving one factor-free odd number.

\\text{Among the three odd terms, one has no factor }3\\text{ or }5

Detailed analysis

There are three odd numbers. At most one is divisible by $3$ , and at most one by $5$ . Therefore one odd number has neither factor $3$ nor factor $5$ ; it also has no factor $2$ . With the restriction above, that number must be $1$ .