Problem
NT-B1-M03-P019 A Cube Is Not 2 Modulo 7
#19
★★★☆☆ Level 3 of 5
Prove that the congruence \(x^3\equiv2\pmod7\) has no solutions.
Make the table of cubes modulo \(7\).
For residues \(0,1,2,3,4,5,6\), the cubes modulo \(7\) are \(0,1,1,6,1,6,6\). Only \(0,1,6\) are possible. Residue \(2\) does not occur, so there are no solutions.
Cubic residues are used less often, but give strong obstructions.