To find the number of numbers that can be made using the digits 2, 4, 5, 6, and 8, with repetition allowed and not exceeding 10,000, we can break down the problem into four cases based on the number of digits in the numbers:

Case 1: 1-digit numbers
In this case, we can simply count the number of digits available, which is 5. Therefore, there are 5 possible 1-digit numbers.

Case 2: 2-digit numbers
In this case, we can use the available digits to form the numbers. Since repetition is allowed, each digit can be used multiple times. However, the numbers cannot exceed 10,000, so the first digit cannot be 0.

- For the first digit, we have 4 choices (2, 4, 5, 6).
- For the second digit, we have 5 choices (including 0).
Therefore, there are 4 x 5 = 20 possible 2-digit numbers.

Case 3: 3-digit numbers
Similar to the previous case, we have 4 choices for the first digit and 5 choices each for the second and third digits. So, there are 4 x 5 x 5 = 100 possible 3-digit numbers.

Case 4: 4-digit numbers
Again, we have 4 choices for the first digit and 5 choices each for the remaining three digits. So, there are 4 x 5 x 5 x 5 = 500 possible 4-digit numbers.

Total number of numbers = Case 1 + Case 2 + Case 3 + Case 4
= 5 + 20 + 100 + 500
= 625

However, the question asks for the numbers not exceeding 10,000. So, we need to subtract the numbers that are greater than 10,000.

Numbers greater than 10,000 can only be in the 4-digit case, where the first digit is 2, 4, 5, or 6. So, we need to subtract the number of 4-digit numbers starting with each of these digits.

For each starting digit, there are 5 choices each for the remaining three digits. So, there are 4 x 5 x 5 x 5 = 500 numbers greater than 10,000.

Therefore, the total number of numbers not exceeding 10,000 is 625 - 500 = 125.

The answer given in the options is 780, which does not match the calculated result. There may be an error in the options provided.