Differences of # and the reverse of # problem 
Millie wrote a five-digit whole number on a blackboard and she also wrote it in reversed order. She considered the difference of her two numbers, and then told Lucy the last three digits of this difference. Can Lucy figure out the first two digits of the difference from this information alone or there could be more than one possibility?

This is an extremely difficult question, and I really have no idea how to solve this!
I came up with the equation $9999A+990B-990D-9999E=Z$ if $Z$ is the difference of $ABCDE$ and $EDCBA$ if $ABCDE$ is the original number! But I cannot solve for $A, B, C, D$ or $E$ with just a single statement.
I feel you can only solve this with logic (and probably computing brute force) e.g. for ### as the last three digits the beginning can be # or #. . .
 A: Recall the tests for divisibility by $9$ and $11$:

*

*The sum of digits of the number is $\equiv\pmod 9$ to the original number

*The sum/difference of the digits of the number (units minus tens plus hundreds minus thousands plus tens of thousands etc.) is $\equiv\pmod{11}$ to the original number.

If the numbers $\overline{ABCDE}$ and $\overline{EDCBA}$ are written using digits $A, B, C, D, E$ in those two orders, then they are both $\equiv\pmod{9}$ to $A+B+C+D+E$ and they are both $\equiv\pmod{11}$ to $A-B+C-D+E$. This means that their difference is divisible by $9$ and by $11$.
Thus, if someone has provided last three digits $\overline{RST}$ of the difference $\overline{PQRST}$, then the whole difference is given by:
$$\begin{array}{rcll}\overline{PQRST}&\equiv&\overline{RST}&\pmod{1000}\\\overline{PQRST}&\equiv&0&\pmod{9}\\\overline{PQRST}&\equiv&0&\pmod{11}\end{array}$$
As per Chinese Remainder Theorem, this determines $\overline{PQRST}$ up to $\pmod{9\cdot 11\cdot 1000}$ (as $9$, $11$ and $1000$ are all coprime). As this difference is a difference of two $5$-digit numbers, it is between $0$ and $99999-10000=89999$, so being uniquely determined up to $\pmod{99000}$ means it is uniquely determined, period.
Answer: yes, knowing the last three digit of the difference, one can recover the first two digits by demanding that the whole $5$-digit difference is divisible by $9$ and $11$.
