How many numbers are there between 1 and 200 which are divisible by 3 but not by 7?

Options

A. a) 38
B. b) 45
C. c) 57
D. d) 66

Answer

To solve the problem of finding how many numbers between 1 and 200 are divisible by 3 but not by 7, we can follow these steps:

### Step 1: Find the total numbers divisible by 3
First, we need to determine how many numbers between 1 and 200 are divisible by 3.

- The smallest number divisible by 3 in this range is 3.
- The largest number divisible by 3 in this range is 198.

To find the count of numbers divisible by 3, we can use the formula for the number of terms in an arithmetic sequence:

\[ \text{Count} = \frac{\text{Largest} - \text{Smallest}}{\text{Common Difference}} + 1 \]

Substituting our values:

\[ \text{Count} = \frac{198 - 3}{3} + 1 = \frac{195}{3} + 1 = 65 + 1 = 66 \]

So, there are **66 numbers** between 1 and 200 that are divisible by 3.

### Step 2: Find the total numbers divisible by both 3 and 7
Next, we need to find how many of these numbers are also divisible by 7. A number that is divisible by both 3 and 7 is divisible by their least common multiple (LCM).

The LCM of 3 and 7 is 21.

- The smallest number divisible by 21 in this range is 21.
- The largest number divisible by 21 in this range is 189.

Using the same formula:

\[ \text{Count} = \frac{189 - 21}{21} + 1 \]

Calculating this:

\[ \text{Count} = \frac{168}{21} + 1 = 8 + 1 = 9 \]

So, there are **9 numbers** between 1 and 200 that are divisible by both 3 and 7.

### Step 3: Subtract the numbers divisible by both 3 and 7 from those divisible by 3
Now, we subtract the numbers that are divisible by both 3 and 7 from the total numbers that are divisible by 3:

\[ \text{Total divisible by 3 but not by 7} = \text{Total divisible by 3} - \text{Total divisible by both 3 and 7} \]

Substituting the values:

\[ \text{Total} = 66 - 9 = 57 \]

Thus, there are **57 numbers** between 1 and 200 that are divisible by 3 but not by 7.

### Final Answer
The final answer is **57**.

---