Introduction:
In this question, we are asked to find the smallest 7-digit number using the digits 5, 0, 6, 1, 7, 2, and 4. We need to arrange these digits in a way that the resulting number is the smallest possible.

Step-by-step solution:

1. Start by arranging the digits in ascending order. This will give us the smallest possible number. The ascending order for the given digits is 0, 1, 2, 4, 5, 6, 7.

2. Since we need a 7-digit number, we can place the smallest digit, 0, in the first position.

3. Now, we have 6 remaining digits: 1, 2, 4, 5, 6, and 7. We need to arrange these digits in the remaining 6 positions.

4. To ensure that the number remains as small as possible, we should place the next smallest digit, 1, in the second position.

5. Now, we have 5 remaining digits: 2, 4, 5, 6, and 7. We need to arrange these digits in the remaining 5 positions.

6. Again, to keep the number as small as possible, we should place the next smallest digit, 2, in the third position.

7. Continuing this process, we can place the remaining digits in the remaining positions in ascending order: 4, 5, 6, and 7.

8. Therefore, the smallest 7-digit number using the given digits is 0124567.

Summary:
To find the smallest 7-digit number using the digits 5, 0, 6, 1, 7, 2, and 4, we need to arrange the digits in ascending order. Starting with the smallest digit, 0, we can place the remaining digits in the remaining positions in ascending order. The resulting number is 0124567.