Problem
NT-B1-M03-P017 Impossibility Modulo 11
#17
★★★☆☆ Level 3 of 5
Prove that \(n^2+n+1\) is not divisible by \(11\) for any integer \(n\).
Check all residues \(n\pmod{11}\).
For \(n=0,1,\ldots,10\), the values of \(n^2+n+1\) modulo \(11\) are \(1,3,7,2,10,9,10,2,7,3,1\). Zero does not occur, so divisibility by \(11\) is impossible.
Shows that a complete table can be a rigorous proof.