To find the sum of all natural numbers from 100 to 300 that are exactly divisible by 4 or 5, we need to follow a step-by-step approach.



We start by identifying the numbers between 100 and 300 that are divisible by 4. The multiples of 4 within this range are 100, 104, 108, 112, and so on. We can observe that these numbers are in an arithmetic sequence with a common difference of 4.



To find the last number divisible by 4 within the given range, we can use the formula:

last_number = first_number + (n - 1) * common_difference

Here, the first number is 100, the common difference is 4, and n represents the number of terms. We can calculate n using the formula:

n = (last_number - first_number) / common_difference + 1

In this case, the last number divisible by 4 is 300. Plugging in the values, we get:

n = (300 - 100) / 4 + 1 = 51

Therefore, there are 51 numbers between 100 and 300 that are divisible by 4.



The sum of an arithmetic series can be calculated using the formula:

sum = (n / 2) * (first_number + last_number)

Plugging in the values, we get:

sum = (51 / 2) * (100 + 300) = 51 * 200 = 10,200

Therefore, the sum of all natural numbers from 100 to 300 that are divisible by 4 is 10,200.



Next, we identify the numbers between 100 and 300 that are divisible by 5. The multiples of 5 within this range are 100, 105, 110, 115, and so on. Again, these numbers form an arithmetic sequence with a common difference of 5.



Using the same approach as in Step 2, we find that the last number divisible by 5 within the given range is 300. The number of terms, n, can be calculated as:

n = (last_number - first_number) / common_difference + 1

Plugging in the values, we get:

n = (300 - 100) / 5 + 1 = 41

Therefore, there are 41 numbers between 100 and 300 that are divisible by 5.



Again, using the formula for the sum of an arithmetic series, we can calculate the sum of numbers divisible by 5:

sum = (n