If $a$, $b$, $c$ are three positive integers such that $a^3+b^3=c^3$ then one of the integer is divisible by $7$ Let on contrary that none of the $a$, $b$, $c$ is divisible by $7$. Then either $a^3\equiv  b^3\pmod{7}$ or $b^3\equiv  c^3\pmod{7}$ or $c^3\equiv  a^3\pmod{7}$.
Now how to go further?

There's 36 combinations of $a$ and $b$ in the integers mod 7, where they're not already $0$. Just try checking them.
