To form an 8-digit number using the digits 1, 2, 3, 4, and 5 with repetition allowed, we need to consider the divisibility rule for 4. According to the rule, for a number to be divisible by 4, the last two digits of the number must be divisible by 4.

Given the digits 1, 2, 3, 4, and 5, we need to find a combination that satisfies this rule and forms an 8-digit number.

To find the answer, we can follow these steps:

1. Consider the last two digits:
- The last digit can be any of the given digits, i.e., 1, 2, 3, 4, or 5.
- The second last digit can also be any of the given digits, i.e., 1, 2, 3, 4, or 5.

2. Count the total number of combinations for the last two digits:
- Since repetition is allowed, there are a total of 5 options for each of the last two digits.
- Therefore, the total number of combinations for the last two digits is 5 * 5 = 25.

3. Check each combination to see if it forms a number divisible by 4:
- For each combination, check if the number formed by the last two digits is divisible by 4.
- If the number is divisible by 4, we can proceed to form the remaining 6 digits of the number.

4. Count the total number of combinations for the remaining 6 digits:
- Since repetition is allowed, there are a total of 5 options for each of the remaining 6 digits.
- Therefore, the total number of combinations for the remaining 6 digits is 5^6 = 15625.

5. Multiply the number of combinations for the last two digits by the number of combinations for the remaining 6 digits:
- 25 * 15625 = 390625

6. Compare the calculated value with the given options:
- a) 31250
- b) 97656
- c) 78125
- d) 97657

The calculated value of 390625 does not match with any of the given options.

Therefore, the correct answer is option 'C' (78125).