7. (4 points) With the number written on the board, one of the following operations is allowed:
1) If there is a digit in the original number that is not 9 and has two neighboring digits greater than 0, you can increase this digit by 1, and decrease the neighboring digits by 1.
2) Subtract 1 from any non-zero digit except the last one, and add 3 to the next digit.
3) Decrease any sufficiently large digit by 7.

If, as a result of any of these operations, there are zeros in the leading positions of the number, they are automatically discarded.

Initially, the number on the board was a hundred eights. In the end, a single-digit number remained. Which one exactly?
Answer: 3

Solution:

The first operation corresponds to subtracting a number of the form $910 \cdot 0$, the second and third - subtracting $70 \cdot 0$. Both these operations do not change the remainder of the original number when divided by 7, since 91 is divisible by 7.

$1001=c d o t 91$, and $888888=888 \cdot 1001$. Therefore, to find the remainder of the original number when divided by 7, the number of eights that are multiples of 6 can be discarded. The remaining number is $8888=8008+875+5=8 \cdot 1001+7 \cdot 125+5$ which gives a remainder of 5.