9. Euler's conjecture was refuted by American mathematicians in 1960, who proved that there exists a positive integer $n$ such that $133^{5}+110^{5}+84^{5}+27^{5}=n^{5}$. Find $n$ when it is satisfied.
9. Clearly $n \geqslant 134$. Now we find the upper bound of $n$.
$$
\begin{aligned}
\because \quad n^{5}= & 133^{5}+110^{5}+84^{5}+27^{5}<133^{5} \\
& +110^{5}+(27+84)^{5}<3(133)^{5} \\
& <\frac{3125}{1024}(133)^{5}=\left(\frac{5}{4}\right)^{5}(133)^{5}, \\
\therefore \quad & n<\frac{5}{4}(133), \text { i.e., } n \leqslant 166 .
\end{aligned}
$$

When a positive integer is raised to the fifth power, the last digit remains unchanged,
$\therefore n$ has the same last digit as $133+110+84+27$. The last digit of the latter is 4,
$$
\begin{array}{l}
\therefore n \text { is one of 134, 144, 154, 164. } \\
\because \quad 133 \equiv 1(\bmod 3), 110 \equiv 2(\bmod 3), \\
84 \equiv 0(\bmod 3), 27 \equiv 0(\bmod 3), \\
\therefore n^{5}=133^{5}+110^{5}+84^{5}+27^{5} \text {. } \\
\equiv 1^{5}+2^{5} \equiv 0(\bmod 3) \text {. } \\
\therefore \quad n=144 \text {. } \\
\end{array}
$$