7.5. Misha and Masha had the same multi-digit integer written in their notebooks, ending in 9876. Masha placed a plus sign between the third and fourth digits from the right, while Misha placed a plus sign between the fourth and fifth digits from the right. To the surprise of the schoolchildren, both resulting sums turned out to be the same. What number was originally written by the schoolchildren? Provide all possible answers and prove that there are no others.
Solution: Let the written number have the form $\overline{x 9876}$, where $x$ is also some natural number. Then Misha got the sum $x+9876$, and Masha got the sum $10 x+9+876$. From the equality $x+9876=10 x+9+876$, we find that $x=999$.

Answer: 9999876 and there is no other number.

Recommendations for checking:

| is in the work | points |
| :--- | :--- |
| the correct answer is given and its uniqueness is proven | 7 points |
| the answer is incorrect only due to an arithmetic error | 6 points |
| the condition is correctly written as a Diophantine equation, but the latter is not solved | 5 points |
| cases of $n$-digit numbers are correctly analyzed for specific natural values of $n$ |  |
| the correct answer is given, and the check that the number 9999876 fits the condition of the problem AND/OR there is an unsuccessful attempt to write the condition of the problem in general as an equation in integers | 2 points |
| the correct answer without justification | 1 point |