Problem
NT-B2-M12-P004 A Quadratic Polynomial Modulo 7
#4
★★☆☆☆ Level 2 of 5
Find all integers \(n\) such that \(7\mid n^2+n+1\).
It is enough to check residues of \(n\) modulo \(7\).
Solve \(n^2+n+1\equiv0\pmod7\). Checking residues gives: for \(n\equiv2\), \(4+2+1=7\), and for \(n\equiv4\), \(16+4+1=21\). No other residues work. The answer is \(n\equiv2\) or \(4\pmod7\).
This connects residue tables with polynomials.