Prove that $6|a+b+c $ if and only if $6|a^3 +b^3+c^3$

Hint:

$$a^3-a=(a-1)a(a+1)$$ is divisible by $6$ being the product of three consecutive integers

$$\implies a^3+b^3+c^3+\cdots\equiv a+b+c+\cdots\pmod6$$