7. (10 points) The last 8 digits of $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$ are
【Analysis】By applying the commutative and associative laws of multiplication to integrate $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$, then organize it into $11111111 \times 111111111111$, we can derive that the last 8 digits of $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$ are 87654321; based on this, solve the problem.

【Solution】Solution: $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$
$$
\begin{array}{l}
=(11 \times 101) \times 1001 \times 10001 \times 1000001 \times 111 \\
=(1111 \times 10001) \times 1001 \times 1000001 \times 111 \\
=11111111 \times 111111 \times 1000001 \\
=11111111 \times 111111111111 ;
\end{array}
$$

Since $1111 \times 1111=1234321, 1111 \times 111111=123444321$,
$$
111111 \times 111111=12345654321, \quad 111111 \times 11111111=1234566654321,
$$

Therefore: $11111111 \times 111111111111=1234567888887654321$,
So the last 8 digits of $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$ are 87654321; Answer: The last 8 digits of $11 \times 101 \times 1001 \times 10001 \times 1000001 \times 111$ are 87654321. The answer is: 87654321.