<s>[INST] Solve in positive integers the equation $a^3+b^3=9ab$ Solve in positive integers the equation:
$$a^3+b^3=9ab$$
I try to:
$$\dfrac{a^2}{b}+\dfrac{b^2}{a}=9\Longrightarrow a^2<9b,b^2<9a$$
Of course, I can't solve it. Can anyone help?
 [/INST] By the Arithmetic Mean Geometric Mean Inequality, we have
$$\frac{9}{2}=\frac{1}{2}\left(\frac{a^2}{b}+\frac{b^2}{a}\right)\ge \sqrt{ab}.$$
Now we only have a smallish number of possibilities to examine. 
A simple congruential observation then dramatically cuts down the number of cases. For $a$ and $b$ must both be even. (If they are both odd, the left-hand side is even and the right-hand side is odd. If they are of different parity, the left-hand side is odd and the right-hand side is even.)
Thus $a^3+b^3$ is divisible by $8$, and one of $a$ and $b$ is divisible by $2$, and the other is divisible by $4$. 
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