How many numbers are there in the set `S={200,201,202,….,800}` which are divisible by neither of 5 or 7?

Options

A. 411
B. 412
C. 410
D. None of these

Answer

To solve the problem of finding how many numbers in the set \( S = \{200, 201, 202, \ldots, 800\} \) are divisible by neither 5 nor 7, we can follow these steps:

### Step 1: Calculate the total numbers in the set \( S \)

The total numbers in the set can be calculated as follows:
\[ \text{Total numbers} = 800 - 200 + 1 = 601 \]

### Step 2: Calculate the numbers divisible by 5

To find the numbers divisible by 5 from 200 to 800, we first find the count of numbers divisible by 5 from 1 to 800 and then subtract the count of numbers divisible by 5 from 1 to 199.

- Count of numbers divisible by 5 from 1 to 800:
\[ \left\lfloor \frac{800}{5} \right\rfloor = 160 \]
- Count of numbers divisible by 5 from 1 to 199:
\[ \left\lfloor \frac{199}{5} \right\rfloor = 39 \]
- Therefore, the count of numbers divisible by 5 from 200 to 800:
\[ 160 - 39 = 121 \]

### Step 3: Calculate the numbers divisible by 7

Next, we find the count of numbers divisible by 7 in a similar manner.

- Count of numbers divisible by 7 from 1 to 800:
\[ \left\lfloor \frac{800}{7} \right\rfloor = 114 \]
- Count of numbers divisible by 7 from 1 to 199:
\[ \left\lfloor \frac{199}{7} \right\rfloor = 28 \]
- Therefore, the count of numbers divisible by 7 from 200 to 800:
\[ 114 - 28 = 86 \]

### Step 4: Calculate the numbers divisible by both 5 and 7 (i.e., 35)

Now, we need to find the count of numbers that are divisible by both 5 and 7, which means they are divisible by 35.

- Count of numbers divisible by 35 from 1 to 800:
\[ \left\lfloor \frac{800}{35} \right\rfloor = 22 \]
- Count of numbers divisible by 35 from 1 to 199:
\[ \left\lfloor \frac{199}{35} \right\rfloor = 5 \]
- Therefore, the count of numbers divisible by 35 from 200 to 800:
\[ 22 - 5 = 17 \]

### Step 5: Apply the principle of inclusion-exclusion

Using the principle of inclusion-exclusion, we can find the total count of numbers divisible by either 5 or 7:
\[ \text{Count divisible by 5 or 7} = \text{Count divisible by 5} + \text{Count divisible by 7} - \text{Count divisible by both} \]
\[ = 121 + 86 - 17 = 190 \]

### Step 6: Calculate the numbers not divisible by 5 or 7

Finally, we can find the count of numbers that are not divisible by either 5 or 7:
\[ \text{Count not divisible by 5 or 7} = \text{Total numbers} - \text{Count divisible by 5 or 7} \]
\[ = 601 - 190 = 411 \]

### Final Answer

Thus, the number of numbers in the set \( S \) that are divisible by neither 5 nor 7 is:
\[ \boxed{411} \]