4. (20 points) Before the math lesson, the teacher wrote nine consecutive numbers on the board, but the duty students accidentally erased one of them. When the lesson began, it turned out that the sum of the remaining eight numbers is 1703. Which number did the duty students erase?
Answer: 214.

Solution: Let the average of the original numbers be $a$. Then these numbers can be written in a symmetric form:

$$
a-4, a-3, a-2, a-1, a, a+1, a+2, a+3, a+4
$$

The erased number has the form $a+b$, where $-4 \leqslant b \leqslant 4$, and the sum of the remaining numbers is $9a - (a+b) = 8a - b$. On the other hand, this sum is also equal to $1703 = 8 \cdot 213 - 1$. Therefore,

$$
8 \cdot 213 - 1 = 8a - b, \quad \text{from which} \quad 8(a - 213) = b - 1
$$

Then the number $b-1$ is divisible by 8 and lies in the interval $[-5, 3]$. Thus, $b=1, a=213$, and the number that was erased is $213 + 1 = 214$.