Problem
NT-B1-M12-P013 Set 3. Divisibility of a Quadratic Expression
#13
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Find all \(n\pmod3\) for which \(3\mid n^2+n+1\).
Check residues \(0,1,2\).
For \(n\equiv0\), we get \(1\); for \(n\equiv1\), we get \(3\equiv0\); for \(n\equiv2\), we get \(7\equiv1\pmod3\). Thus only \(n\equiv1\pmod3\) works.
Standard residue table.