Let x, y, z be integers. Find all triples (x, y, z) such that
x^3 + y^3 + z^3 \equiv 0 \pmod{3}
and
x + y + z \equiv 1 \pmod{3}.
From Fermat’s Little Theorem:
a^3 \equiv a \pmod{3}
thus x^3+y^3+z^3 \equiv x+y+z \pmod{3}
But x^3+y^3+z^3 \equiv 0 \pmod{3} and x+y+z \equiv 1 \pmod{3}
Therefore 0 \equiv 1 \pmod{3} (!)
Thus there is no solution
Fabulous