The sum of all three digit numbers divisible by 5 is :

Question

Accepted Answer

To determine the sum of all three-digit numbers divisible by 5, we need to identify the sequence of numbers that meet this criterion and then calculate their sum.

1. **Identify the Sequence:**
   - The smallest three-digit number divisible by 5 is 100.
   - The largest three-digit number divisible by 5 is 995.
   - These numbers form an arithmetic sequence where the first term $ a = 100 $, the last term $ l = 995 $, and the common difference $ d = 5 $.

2. **Calculate the Number of Terms:**
   - The formula for the $ n $-th term of an arithmetic sequence is given by:
     $$
     l = a + (n-1) \times d
     $$
   - Plugging in the known values:
     $$
     995 = 100 + (n-1) \times 5
     $$
     $$
     895 = (n-1) \times 5
     $$
     $$
     n-1 = 179
     $$
     $$
     n = 180
     $$
   - There are 180 terms in this sequence.

3. **Calculate the Sum of the Sequence:**
   - The sum $ S $ of an arithmetic sequence is given by:
     $$
     S = \frac{n}{2} \times (a + l)
     $$
   - Substituting the known values:
     $$
     S = \frac{180}{2} \times (100 + 995)
     $$
     $$
     S = 90 \times 1095
     $$
     $$
     S = 98550
     $$

Thus, the sum of all three-digit numbers divisible by 5 is 98550, which corresponds to Option 4.

**Explanation for Incorrect Options:**

- **Option 1 (98560):** This option is incorrect because the calculated sum of 98550 does not match 98560. The arithmetic sequence and calculations clearly show that the sum is 98550.
  
- **Option 2 (98540):** This option is incorrect because it is less than the calculated sum of 98550. The arithmetic sequence and calculations confirm that the sum is 98550.
  
- **Option 3 (98520):** This option is incorrect because it is significantly less than the calculated sum of 98550. The arithmetic sequence and calculations confirm that the sum is 98550.

In conclusion, the correct answer is Option 4 (98550), as supported by the detailed calculation of the arithmetic sequence of three-digit numbers divisible by 5.