The number of numbers which are multiples of both 3 and 5 in the first 100 natural numbers is:

Options

A. 10
B. 9
C. 7
D. 6

Answer

To find the number of natural numbers that are multiples of both 3 and 5 in the first 100 natural numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Range**: We are looking at the first 100 natural numbers, which are from 1 to 100.

2. **Find the LCM of 3 and 5**:
- The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both.
- The multiples of 3 are: 3, 6, 9, 12, 15, 18, ..., and so on.
- The multiples of 5 are: 5, 10, 15, 20, 25, ..., and so on.
- The LCM of 3 and 5 can be calculated as follows:
- 3 = 3 × 1
- 5 = 5 × 1
- Therefore, LCM(3, 5) = 3 × 5 = 15.

3. **List the Multiples of 15 up to 100**:
- Now, we need to find how many multiples of 15 are there from 1 to 100.
- The multiples of 15 are: 15, 30, 45, 60, 75, 90, 105, ...
- We stop at 100, so we only consider: 15, 30, 45, 60, 75, and 90.

4. **Count the Multiples**:
- The multiples of 15 that are less than or equal to 100 are: 15, 30, 45, 60, 75, and 90.
- There are 6 multiples of 15 in this range.

5. **Conclusion**:
- Therefore, the number of natural numbers which are multiples of both 3 and 5 in the first 100 natural numbers is **6**.

### Final Answer:
The number of numbers which are multiples of both 3 and 5 in the first 100 natural numbers is **6**.