To find the smallest 8-digit number that has four different digits, we can follow these steps:
### Step-by-Step Solution:
1. **Identify the smallest digits**: The smallest digits we can use are 0, 1, 2, and 3. However, since we need to create an 8-digit number, we must consider how to arrange these digits.
2. **Start with the smallest non-zero digit**: Since we are forming an 8-digit number, we cannot start with 0 (as it would not count as an 8-digit number). Therefore, we will start with the smallest non-zero digit, which is 1.
3. **Fill in the remaining digits**: After placing 1 at the beginning, we need to fill in the remaining digits. We have already used 1, and we still need to use 0, 2, and 3.
4. **Construct the number**: To make the number as small as possible, we will use 0 as many times as we can after 1. Therefore, we will place five 0s after 1. This gives us:
- 1 (first digit)
- 0, 0, 0, 0, 0 (next five digits)
5. **Add the remaining digits**: Now, we need to add the last two digits. We will add 2 and 3. To keep the number small, we will place 2 before 3.
6. **Final arrangement**: Putting it all together, we get:
- 1 (first digit)
- 0, 0, 0, 0, 0 (next five digits)
- 2 (seventh digit)
- 3 (eighth digit)
Thus, the smallest 8-digit number having four different digits is **10000023**.
### Summary of the Solution:
The smallest 8-digit number having four different digits is **10000023**.
