To solve the problem, we need to find all natural numbers \( n \) such that \( 100 < n < 200 \) and \( \text{HCF}(91, n) > 1 \).
### Step-by-Step Solution:
1. **Identify the Factors of 91**:
- First, we need to find the factors of 91. The prime factorization of 91 is:
\[
91 = 7 \times 13
\]
- Therefore, the factors of 91 are \( 1, 7, 13, 91 \).
2. **Determine the Condition for HCF**:
- For \( \text{HCF}(91, n) > 1 \), \( n \) must be a multiple of either 7 or 13 (since these are the factors of 91 greater than 1).
3. **Find Multiples of 7**:
- The multiples of 7 between 100 and 200 can be found by calculating:
- The smallest multiple of 7 greater than 100:
\[
7 \times 15 = 105
\]
- The largest multiple of 7 less than 200:
\[
7 \times 28 = 196
\]
- The multiples of 7 in this range are:
\[
105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196
\]
4. **Find Multiples of 13**:
- The multiples of 13 between 100 and 200 can be found by calculating:
- The smallest multiple of 13 greater than 100:
\[
13 \times 8 = 104
\]
- The largest multiple of 13 less than 200:
\[
13 \times 15 = 195
\]
- The multiples of 13 in this range are:
\[
104, 117, 130, 143, 156, 169, 182, 195
\]
5. **Combine the Results**:
- Now we combine the multiples of 7 and 13, ensuring we do not double count:
- From multiples of 7: \( 105, 112, 119, 126, 133, 140, 147, 154, 161, 168, 175, 182, 189, 196 \)
- From multiples of 13: \( 104, 117, 130, 143, 156, 169, 182, 195 \)
- The combined list (removing duplicates) is:
\[
104, 105, 112, 117, 119, 126, 130, 133, 140, 143, 147, 154, 156, 161, 168, 169, 175, 182, 189, 195, 196
\]
6. **Sum the Valid Numbers**:
- Now, we sum all the unique numbers from the combined list:
\[
104 + 105 + 112 + 117 + 119 + 126 + 130 + 133 + 140 + 143 + 147 + 154 + 156 + 161 + 168 + 169 + 175 + 182 + 189 + 195 + 196
\]
- Performing the addition gives:
\[
= 104 + 105 + 112 + 117 + 119 + 126 + 130 + 133 + 140 + 143 + 147 + 154 + 156 + 161 + 168 + 169 + 175 + 182 + 189 + 195 + 196 = 3320
\]
### Final Answer:
The sum of all natural numbers \( n \) such that \( 100 < n < 200 \) and \( \text{HCF}(91, n) > 1 \) is \( 3320 \).
