3 Let positive integers $a, b$ satisfy $a^{2} b \mid\left(a^{3}+b^{3}\right)$, prove $a=b$.

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3. Let $a^{3}+b^{3}=m a^{2} b$, then $\left(\frac{a}{b}\right)^{3}-m\left(\frac{a}{b}\right)^{2}+1=0$, which means the rational number $\frac{a}{b}$ is a root of the integral coefficient equation with leading coefficient 1:
$$x^{3}-m x^{2}+1=0$$

Therefore, $\frac{a}{b}$ must be an integer (Exercise 2, Question 4). On the other hand, any integer root of equation (1) must divide the constant term 1, and thus can only be $\pm 1$; since $a, b$ are positive, we have $\frac{a}{b}=1$, i.e., $a=b$.