20 Find the smallest positive integer $a$, such that for any integer $x$, we have $65 \mid\left(5 x^{13}+13 x^{5}+9 a x\right)$.
20. From the condition, we know $65 \mid (18 + 9a)$ (take $x=1$), and since $(9,65)=1$, it follows that $65 \mid a+2$, hence $a \geq 63$.

When $a=63$, using Fermat's Little Theorem, we know that for any integer $x$, we have
$$\begin{aligned}
& 5 x^{13} + 13 x^{5} + 9 a x \\
\equiv & 13 x + 9 a x \\
\equiv & (3 + (-1) \times 3) x \\
\equiv & 0 \pmod{5}
\end{aligned}$$
$$\begin{aligned}
& 5 x^{13} + 13 x^{5} + 9 a x \\
\equiv & 5 x + 9 a x \\
\equiv & (5 + 9 \times (-2)) x \\
\equiv & 0 \pmod{13}
\end{aligned}$$

Therefore,
$$65 \mid 5 x^{13} + 13 x^{5} + 9 a x$$

In summary, the smallest positive integer $a=63$.