Five-digits smallest number, which is completely divisible by 75 is

Options

A. 10025
B. 10050
C. 10075
D. 9975

Answer

To find the smallest five-digit number that is completely divisible by 75, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the smallest five-digit number**:
The smallest five-digit number is 10,000.

2. **Check divisibility by 75**:
A number is divisible by 75 if it is divisible by both 3 and 25.

3. **Check divisibility by 25**:
A number is divisible by 25 if its last two digits form a number that is divisible by 25. The last two digits of 10,000 are 00, which is divisible by 25.

4. **Check divisibility by 3**:
A number is divisible by 3 if the sum of its digits is divisible by 3. The sum of the digits of 10,000 is 1 + 0 + 0 + 0 + 0 = 1, which is not divisible by 3.

5. **Find the next number that is divisible by 75**:
Since 10,000 is not divisible by 3, we need to find the next number that is divisible by both 3 and 25.
- We can do this by adding numbers to 10,000 until we find one that meets both criteria.
- The next number to check is 10,000 + 1 = 10,001, then 10,002, and so on.

6. **Check 10,050**:
- 10,050 is the next candidate.
- Check divisibility by 25: The last two digits are 50, which is divisible by 25.
- Check divisibility by 3: The sum of the digits of 10,050 is 1 + 0 + 0 + 5 + 0 = 6, which is divisible by 3.

7. **Conclusion**:
Since 10,050 is divisible by both 3 and 25, it is also divisible by 75. Therefore, the smallest five-digit number that is completely divisible by 75 is **10,050**.

### Final Answer:
The smallest five-digit number that is completely divisible by 75 is **10,050**.