MathLabs

Problem 4

Find all positive integers nn such that {n,n+1,n+2,n+3,n+4,n+5}\{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 residue 00 modulo 77
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In plain words

A single multiple of 77 cannot be shared by both products.

7midm for some min{n,ldots,n+5} is impossible7\\mid m\text{ for some }m\\in\{n,\\ldots,n+5\}\text{ is impossible}
Detailed analysis

Among six consecutive integers at most one is divisible by 77. If so, the factor 77 would divide only one side of any partition, contradicting equal products.