10. $1991^{2000}$ divided by $10^{6}$, the remainder is
$$
\begin{array}{l}
1991^{2000}=(1990+1)^{2000} \equiv C_{2000}^{2} 1990^{2}+C_{2000}^{1} 1990+1 \\
=1999 \cdot 199^{2} \cdot 10^{5}+199 \cdot 2 \cdot 10^{4}+1=199 \cdot 10^{4}(1999 \cdot 1990+2)+1 \\
=199 \cdot 3978012 \cdot 10^{4}+1 \equiv 199 \cdot 12 \cdot 10^{4}+1 \equiv 880001\left(\bmod 10^{6}\right) .
\end{array}
$$

The translation is as follows:

$$
\begin{array}{l}
1991^{2000}=(1990+1)^{2000} \equiv C_{2000}^{2} 1990^{2}+C_{2000}^{1} 1990+1 \\
=1999 \cdot 199^{2} \cdot 10^{5}+199 \cdot 2 \cdot 10^{4}+1=199 \cdot 10^{4}(1999 \cdot 1990+2)+1 \\
=199 \cdot 3978012 \cdot 10^{4}+1 \equiv 199 \cdot 12 \cdot 10^{4}+1 \equiv 880001\left(\bmod 10^{6}\right) .
\end{array}
$$