18. Find the number of positive integer pairs $(a, b)$ satisfying $a^{2}+b^{2}<2013$ and $a^{2} b \mid\left(b^{3}-a^{3}\right)$.
18. Answer: 31 .
Let $k=\frac{b^{3}-a^{3}}{a^{2} b}=\left(\frac{b}{a}\right)^{2}-\frac{a}{b}$. Then
$$
\left(\frac{a}{b}\right)^{3}+k\left(\frac{a}{b}\right)^{2}-1=0 .
$$

The only possible positive rational solution of $x^{3}+k x^{2}-1=0$ is $x=1$; namely, $a=b$. Conversely, if $a=b$, then it is obvious that $a^{2} b \mid\left(b^{3}-a^{3}\right)$.

Then $2013>a^{2}+b^{2}=2 a^{2}$ implies $a \leq 31$.