<s>[INST] Show $9 \mid a^3 + b^3 + c^3 \Rightarrow 3 \mid a \text{ or } 3 \mid b \text{ or } 3 \mid c$ Let $a, b,c \in \mathbb{Z}: 9 \mid a^3 + b^3 + c^3$
I need to show that $3 \mid a \text{ or } 3 \mid b \text{ or } 3 \mid c$
I don´t have any clue how should I approach this problem. Could someone give my hints or name some theorems that should I use.
Thank you in advance
 [/INST] If $x\not\equiv 0\pmod{3}$, then $x^3\equiv \pm 1\pmod{9}$, since
$ (3x\pm 1)^3 = 9X\pm 1$.
The sum of three numbers in $\{-1,+1\}$ cannot be $\equiv 0\pmod{9}$, hence $a^3+b^3+c^3\equiv 0\pmod{9}$ implies that at least one number among $a,b,c$ is a multiple of three.
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