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 1 of 4: Exclude large prime factors

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In plain words

A prime occurring only once cannot be balanced between the two products.

p\\ge7 cannot divide any member of a valid block

Detailed analysis

A prime $p\ge7$ divides at most one of six consecutive integers. If it divided one member, its exponent would occur in only one of the two products, so equal products would be impossible. Thus all prime factors are among $2,3,5$ .