9. Given that the largest real root of the equation $x^{3}-7 x^{2}+1=0$ is $t$, then the remainder when $\left[t^{20100}\right]$ is divided by 7
9. Let $f(x)=x^{3}-7 x^{2}+1, f(-1)0, f(1)0$, so the largest real root $t>1$, let the other two roots be $a, \beta$, it is easy to know that $|\alpha|,|\beta|$ are both less than 1, let $g(n)=t^{n}+a^{n}+\beta^{n}$.
So $g(n+3)=7 g(n+2)-g(n)$
So $g(n+6) \equiv g(n)(\bmod 7)$.
So $g(2000) \equiv g(2)(\bmod 7)$.
Since $t^{2000}+\alpha^{2000}+\beta^{2000}=\left[t^{2000}\right]+1$,
and $g(2)=t^{2}+\alpha^{2}+\beta^{2}$
$$
\begin{array}{l}
=(t+\alpha+\beta)^{2}-2(t a+\alpha \beta+\beta t) \\
=7^{2} \equiv 0(\bmod 7)
\end{array}
$$

So $\left[t^{2000}\right] \equiv 6(\bmod 7)$, hence the answer is 6.