In the above question total numbers in the set of numbers `S = {200, 201, 800}` which are either divisible by 5 or by 7 is :

Options

A. 210
B. 190
C. 199
D. can't be determined

Answer

To solve the problem of finding the total numbers in the set \( S = \{200, 201, \ldots, 800\} \) that are either divisible by 5 or by 7, we can follow these steps:

### Step 1: Find numbers divisible by 5
1. **Identify the first and last numbers divisible by 5 in the range.**
- The first number divisible by 5 in the range is 200.
- The last number divisible by 5 in the range is 800.

2. **Use the formula for the number of terms in an arithmetic sequence.**
- The formula is given by:
\[ n = \frac{\text{last term} - \text{first term}}{\text{common difference}} + 1 \]
- Here, the common difference is 5.
- Substituting the values:
\[ n = \frac{800 - 200}{5} + 1 = \frac{600}{5} + 1 = 120 + 1 = 121 \]

### Step 2: Find numbers divisible by 7
1. **Identify the first and last numbers divisible by 7 in the range.**
- The first number divisible by 7 greater than or equal to 200 is 203 (since \( 200 \div 7 \approx 28.57 \), the next whole number is 29, and \( 29 \times 7 = 203 \)).
- The last number divisible by 7 less than or equal to 800 is 798 (since \( 800 \div 7 \approx 114.29 \), the whole number is 114, and \( 114 \times 7 = 798 \)).

2. **Use the formula for the number of terms in an arithmetic sequence.**
- The common difference is 7.
- Substituting the values:
\[ n = \frac{798 - 203}{7} + 1 = \frac{595}{7} + 1 = 85 + 1 = 86 \]

### Step 3: Find numbers divisible by both 5 and 7 (i.e., divisible by 35)
1. **Identify the first and last numbers divisible by 35 in the range.**
- The first number divisible by 35 greater than or equal to 200 is 210 (since \( 200 \div 35 \approx 5.71 \), the next whole number is 6, and \( 6 \times 35 = 210 \)).
- The last number divisible by 35 less than or equal to 800 is 770 (since \( 800 \div 35 \approx 22.86 \), the whole number is 22, and \( 22 \times 35 = 770 \)).

2. **Use the formula for the number of terms in an arithmetic sequence.**
- The common difference is 35.
- Substituting the values:
\[ n = \frac{770 - 210}{35} + 1 = \frac{560}{35} + 1 = 16 + 1 = 17 \]

### Step 4: Apply the principle of inclusion-exclusion
- The total number of integers divisible by either 5 or 7 is given by:
\[ \text{Total} = (\text{Count of numbers divisible by 5}) + (\text{Count of numbers divisible by 7}) - (\text{Count of numbers divisible by both 5 and 7}) \]
- Substituting the values:
\[ \text{Total} = 121 + 86 - 17 = 190 \]

### Final Answer
The total numbers in the set \( S \) that are either divisible by 5 or by 7 is **190**.

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