<s>[INST] Prove that $6 | (a+b+c)$ if and only if $6 | (a^3 + b^3 +c^3).$ I have tried the question but not sure if my solution is correct or not...
My try..
\begin{align}a^3 + b^3 + c^3 = (a+b+c)^3 - 3(a+b)(b+c)(c+a)\end{align}
So , if
\begin{align}6 |(a^3 + b^3 + c^3)\end{align}
Then,
\begin{align}6 | [(a+b+c)^3 - 3(a+b)(b+c)(c+a)]\end{align}
So, also
\begin{align}6 | (a+b+c)^3\end{align}
\begin{align}6 | (a+b+c)(a+b+c)(a+b+c)\end{align}
So,
\begin{align}6|(a+b+c)\end{align}
Is my solution correct ?
 [/INST] $a^3+b^3+c^3=(a+b+c)^3-3(a+b)(b+c)(a+c)$
At least one pair from $(a+b) , (b+c)  , (a+c)$ is even number. 
Therefore, $6 | 3(a+b)(b+c)(a+c)$. Consequently, 
$a^3+b^3+c^3 \equiv (a+b+c)^3 \mod{6}$
$a^3+b^3+c^3 \equiv a+b+c \mod{6}$
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