Solution:
In order to find the number of numbers between 74 and 25556 that are divisible by 5, we need to find how many numbers are there between these two numbers that are divisible by 5.

Method 1: Using division

- Divide 74 by 5, we get 14 as the quotient and 4 as the remainder.
- The next multiple of 5 after 74 is 75, which is divisible by 5.
- Divide 25556 by 5, we get 5111 as the quotient and 1 as the remainder.
- The previous multiple of 5 before 25556 is 25555, which is divisible by 5.
- Now, we need to count the number of multiples of 5 between 75 and 25555.
- To do this, we can subtract the two multiples of 5, and then divide the result by 5 and add 1.
- So, the number of multiples of 5 between 75 and 25555 is ((25555 - 75) / 5) + 1 = 5097.

Method 2: Using floor division

- We can also use floor division to find the number of multiples of 5 between 74 and 25556.
- Divide 74 by 5 using floor division, we get 14.
- Divide 25556 by 5 using floor division, we get 5111.
- The number of multiples of 5 between 74 and 25556 is (5111 - 14) + 1 = 5098.
- However, we need to exclude 74 from this count since it is not divisible by 5.
- So, the final answer is 5098 - 1 = 5097.

Hence, the correct answer is option B, 5097.