The numbers of natural numbers `lt 300` that are divisible by 6 but not by 9 :

Options

A. 49
B. 37
C. 33
D. 16

Answer

To solve the problem of finding the number of natural numbers less than 300 that are divisible by 6 but not by 9, we can follow these steps:

### Step 1: Find the count of natural numbers less than 300 that are divisible by 6.

To find the count of numbers divisible by 6, we can use the formula for the nth term of an arithmetic progression (AP):

\[ n = \frac{\text{Last term} - \text{First term}}{\text{Common difference}} + 1 \]

Here, the first term \(A\) is 6, the common difference \(d\) is 6, and the last term is the largest multiple of 6 less than 300, which is 294.

So, we set up the equation:

\[ n = \frac{294 - 6}{6} + 1 \]

Calculating this gives:

\[ n = \frac{288}{6} + 1 = 48 + 1 = 49 \]

Thus, there are **49 natural numbers less than 300 that are divisible by 6**.

### Step 2: Find the count of natural numbers less than 300 that are divisible by both 6 and 9 (i.e., divisible by 18).

Next, we find the count of numbers divisible by 18. Again, we use the formula for the nth term of an AP:

Here, the first term \(A\) is 18, the common difference \(d\) is 18, and the last term is the largest multiple of 18 less than 300, which is 288.

So, we set up the equation:

\[ n = \frac{288 - 18}{18} + 1 \]

Calculating this gives:

\[ n = \frac{270}{18} + 1 = 15 + 1 = 16 \]

Thus, there are **16 natural numbers less than 300 that are divisible by 18**.

### Step 3: Subtract the count of numbers divisible by 18 from the count of numbers divisible by 6.

Now, we subtract the count of numbers divisible by 18 from the count of numbers divisible by 6 to find those that are divisible by 6 but not by 9:

\[ \text{Count of numbers divisible by 6 but not by 9} = 49 - 16 = 33 \]

Thus, the final answer is that there are **33 natural numbers less than 300 that are divisible by 6 but not by 9**.

### Summary of Steps:
1. Count numbers divisible by 6: 49
2. Count numbers divisible by 18: 16
3. Subtract to find numbers divisible by 6 but not by 9: 33