Prove that $7 \mid abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$

Question

Let $a,b,c$ be positive integer. Prove that $abc(a^3-b^3)(b^3-c^3)(c^3-a^3)$ is divisible by $7$.

Answer 1

Hint: If $x\not\equiv 0\pmod{7}$, then $x^3\equiv \pm 1\pmod{7}$.