Which of the following are prime numbers ?
(i) 103 (ii) 137 (iii) 161 (iv) 179 (v) 217 (vi) 277 (vii) 331 (viii) 397

Answer

To determine which of the given numbers are prime numbers, we will check each number to see if it has only two distinct positive factors: 1 and itself.

**Step-by-Step Solution:**

1. **Check 103:**
- The factors of 103 are 1 and 103.
- Since it has only two factors, **103 is a prime number.**

2. **Check 137:**
- The factors of 137 are 1 and 137.
- Since it has only two factors, **137 is a prime number.**

3. **Check 161:**
- The factors of 161 are 1, 7, 23, and 161.
- Since it has more than two factors, **161 is not a prime number.**

4. **Check 179:**
- The factors of 179 are 1 and 179.
- Since it has only two factors, **179 is a prime number.**

5. **Check 217:**
- The factors of 217 are 1, 7, 31, and 217.
- Since it has more than two factors, **217 is not a prime number.**

6. **Check 277:**
- The factors of 277 are 1 and 277.
- Since it has only two factors, **277 is a prime number.**

7. **Check 331:**
- The factors of 331 are 1 and 331.
- Since it has only two factors, **331 is a prime number.**

8. **Check 397:**
- The factors of 397 are 1 and 397.
- Since it has only two factors, **397 is a prime number.**

**Final List of Prime Numbers:**
- Prime numbers from the list are: **103, 137, 179, 277, 331, and 397.**
- Non-prime numbers from the list are: **161 and 217.**

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