11. A student missed the multiplication sign written between two three-digit numbers, mistaking them for a six-digit number, which turned out to be exactly 7 times the original product. What is the sum of these two three-digit numbers?
【Analysis】Let these two three-digit numbers be $\mathrm{X}, \mathrm{Y}$, then according to the problem, we have:
$$
1000 \mathrm{X}+\mathrm{Y}=7 \mathrm{XY}
$$

Then $49 \mathrm{XY} - 7000 \mathrm{X} - 7 \mathrm{Y} + 1000 = 1000$
That is, $(7 \mathrm{X}-1)(7 \mathrm{Y}-1000)=1000$
Since $\mathrm{X}, \mathrm{Y}$ are both three-digit integers,
it follows that $7 \mathrm{X}-1$ is at least 699
and 1000 has only one divisor greater than 699, which is 1000 itself.
Therefore, $7 \mathrm{X}-1=1000, 7 \mathrm{Y}-1000=1$
Solving these, we get $\mathrm{X}=\mathrm{Y}=143$
Thus, the sum of the two numbers is 286