<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] Fact is that, $n^3−n=(n−1)n(n+1)$ being a product of three consecutive integers is a multiple of $3$. 
Hence $3$ divides $(a^3−a)+(b^3−b)+(c^3−c)$ for any triple of integers $a,b,c$. Rearranging the terms, we have that $3$ dividing $(a^3+b^3+c^3)−(a+b+c)$.
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