<s>[INST] Given that $a$ and $b$ are integers satisfied $3 \mid ab(a + b) + 2$, prove that $9 \mid ab(a + b) + 2$. 
Given that $a$ and $b$ are integers satisfy $3 \mid [ab(a + b) + 2]$, prove that $9 \mid [ab(a + b) + 2]$.

This problem is adapted from a recent competition. (The last few ones were and the next few ones will be.) I want to ask if the solution I have provided below is correct and if there are any other solutions which are more suitable more a test setting.
 [/INST] Clearly $3\nmid a,b$. Fermat little theorem we have $a^3\equiv_3 a$ so multiplying given relation with $a$ on the right we get $$3\mid a^2b^2+ab+2a\implies 3\mid 4a^2b^2+4ab+8a$$
$$\implies 3\mid (2ab+1)^2+8a-1\implies a+1\equiv_3 (2ab+1)^2$$ 
Since $(2ab+1)^2 \equiv_3 0,1$ we see, since $3\nmid a$ that $a\equiv_3 -1$. The same is true for $b\equiv_3 -1$. So $a=3x-1$ and $b=3y-1$. Now we have:
$$ (3x-1)^2(3y-1)+(3y-1)^2(3x-1)+2\equiv_9$$ $$\equiv_9(3x+1)(3y-1)+(3y+1)(3x-1)+2\equiv_9  0$$
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