$2.1999^{2000}$ divided by $10^{10}$, the remainder is $\qquad$
$$
\begin{array}{l}
2.5996000001 . \\
1999^{2000}=(1-2000)^{2000} \\
=1-2000 \times 2000+\frac{1}{2} \times 2000 \times 1999 \times 2000^{2} \\
\quad-\frac{1}{16} \times 2000 \times 1999 \times 1998 \times 2000^{3}+\cdots \\
\quad+2000^{2000} .
\end{array}
$$

In the expression on the right side of the equation above, each term from the fourth term onward can be divided by $10^{10}$, so the remainder when $1999^{2000}$ is divided by $10^{10}$ is the remainder when the first three terms of the expression on the right side are divided by $10^{10}$. It is easy to calculate that this remainder is 5996000001.