The number of times the digit 8 will be written when listing the integers from 1 to 1000 is :

Options

A. 100
B. 200
C. 300
D. 400

Answer

To find the number of times the digit 8 appears when listing the integers from 1 to 1000, we can break down the problem into manageable steps.

### Step-by-Step Solution:

1. **Count occurrences of 8 in the units place (1 to 999)**:
- The numbers with 8 in the units place from 1 to 999 are: 8, 18, 28, 38, 48, 58, 68, 78, 88, 98, ..., 998.
- These numbers form an arithmetic sequence where the first term is 8 and the common difference is 10.
- The last term in this sequence is 998.
- To find the number of terms (n) in this sequence, we can use the formula for the nth term of an arithmetic sequence:
\[ a_n = a + (n-1)d \]
where \(a\) is the first term, \(d\) is the common difference, and \(a_n\) is the last term.
\[ 998 = 8 + (n-1) \times 10 \]
\[ 990 = (n-1) \times 10 \]
\[ n-1 = 99 \implies n = 100 \]
- Therefore, there are 100 occurrences of the digit 8 in the units place.

2. **Count occurrences of 8 in the tens place (1 to 999)**:
- The numbers with 8 in the tens place from 1 to 999 are: 80-89, 180-189, 280-289, ..., 980-989.
- Each complete set of hundreds contributes 10 occurrences of 8 in the tens place.
- There are 10 complete sets of hundreds from 0 to 900 (0-99, 100-199, ..., 900-999).
- Thus, the total occurrences in the tens place is:
\[ 10 \text{ (from each hundred)} \times 10 \text{ (hundreds)} = 100 \]

3. **Count occurrences of 8 in the hundreds place (1 to 999)**:
- The only numbers with 8 in the hundreds place from 1 to 999 are from 800 to 899.
- This gives us 100 occurrences (800, 801, ..., 899).
- Therefore, there are 100 occurrences of the digit 8 in the hundreds place.

4. **Combine all occurrences**:
- Total occurrences of the digit 8 from 1 to 999:
\[ \text{Units place} + \text{Tens place} + \text{Hundreds place} = 100 + 100 + 100 = 300 \]

5. **Consider the number 1000**:
- The number 1000 does not contain the digit 8.
- Therefore, the total count remains the same.

### Final Answer:
The total number of times the digit 8 will be written when listing the integers from 1 to 1000 is **300**.