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 4 of 4: Use quadratic residues
Previous stepNext step
In plain words

The total product would have to be a square, but its residue is a nonsquare.

P2equiv−1equiv6pmod7isimpossibleP^2\\equiv-1\\equiv6\\pmod7 is impossible
Detailed analysis

If the two products were equal to PP, the total product would be P2P^2. But the squares modulo 77 are 0,1,2,40,1,2,4, not 6=−16=-1. Contradiction.