<s>[INST] $3\mid a^3+b^3+c^3$ if only if $3\mid a+b+c $ 
Prove the following equivalence: $3\mid (a^3 + b^3 + c^3) $ if and only if $3\mid (a + b + c) $.

My try:
I know $a^3 + b^3 + c^3 = (a + b + c) (a^2 + b^2 + c^2 – ab – bc – ca) + 3abc$, but I can't seem to proceed from here.
Thanks all!
 [/INST] *

*$1^3 = 1 \mod 3$

*$2^3 = 2 \mod 3$

*$3^3 = 3 \mod 3$


This means $x = x^3 \mod 3$ for any integer $x$. This also means that if $a^3 + b^3 + c^3 = 0 \mod 3$ then $a + b + c = 0 \mod 3$.
Thus it is divisible by three.
</s>