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 2 of 4: List the nonzero residues
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In plain words

Consecutive numbers cycle through every residue class.

{n,n+1,ldots,n+5}equiv{1,2,3,4,5,6}pmod7\{n,n+1,\\ldots,n+5\}\\equiv\{1,2,3,4,5,6\}\\pmod7
Detailed analysis

With residue 00 excluded, the six consecutive numbers represent all six nonzero residue classes modulo 77.