Let p,q,r be natural numbers .if m is their LCM and n their HCF , consider the following :
1. mn = pqr if each one of p,q,r is prime
2. mn = pqr, if , p, q, r are relatively prime in pairs .

Options

A. 1 only
B. 2 only
C. both 1 and 2
D. neither 1 nor 2

Answer

To solve the problem, we need to analyze the two statements given regarding the natural numbers \( p, q, r \) and their LCM (Least Common Multiple) \( m \) and HCF (Highest Common Factor) \( n \).

### Step-by-Step Solution

#### Part 1: When \( p, q, r \) are prime numbers

1. **Choose prime numbers for \( p, q, r \)**:
- Let \( p = 2 \), \( q = 3 \), and \( r = 5 \).

2. **Calculate the HCF of \( p, q, r \)**:
- Since all three numbers are prime and have no common factors other than 1, the HCF \( n = 1 \).

3. **Calculate the LCM of \( p, q, r \)**:
- The LCM of prime numbers is simply their product:
\[ m = p \times q \times r = 2 \times 3 \times 5 = 30. \]

4. **Verify the relationship \( mn = pqr \)**:
- Calculate \( mn \):
\[ mn = 30 \times 1 = 30. \]
- Calculate \( pqr \):
\[ pqr = 2 \times 3 \times 5 = 30. \]
- Since \( mn = pqr \), the first statement is true.

#### Part 2: When \( p, q, r \) are relatively prime in pairs

1. **Choose numbers for \( p, q, r \)**:
- Let \( p = 4 \), \( q = 9 \), and \( r = 25 \). (These numbers are relatively prime in pairs).

2. **Calculate the HCF of \( p, q, r \)**:
- Since \( 4 \), \( 9 \), and \( 25 \) have no common factors, the HCF \( n = 1 \).

3. **Calculate the LCM of \( p, q, r \)**:
- The LCM of \( p, q, r \) can be calculated as:
\[ m = LCM(4, 9, 25) = 4 \times 9 \times 25 = 900. \]

4. **Verify the relationship \( mn = pqr \)**:
- Calculate \( mn \):
\[ mn = 900 \times 1 = 900. \]
- Calculate \( pqr \):
\[ pqr = 4 \times 9 \times 25 = 900. \]
- Since \( mn = pqr \), the second statement is also true.

### Conclusion
Both statements are true. Therefore, the correct answer is that both conditions hold.