Problem
NT-B1-M03-P012 Sum of Three Cubes Modulo 9
#12
★★☆☆☆ Level 2 of 5
Prove that a sum of three integer cubes cannot have residue \(4\) or \(5\) modulo \(9\).
A cube modulo \(9\) is \(0\), \(1\), or \(-1\).
Each cube modulo \(9\) is \(0,\pm1\). A sum of three such residues is among \(-3,-2,-1,0,1,2,3\), i.e. modulo \(9\) among \(6,7,8,0,1,2,3\). Residues \(4\) and \(5\) do not occur.
A very useful table for stronger problems about cubes.