If the product of the HCF and the LCM of 3 natural numbers p,q,r equals pqr, then p,q,r must be :

Options

A. such that `(p,q,r) = 1`
B. prime number
C. odd number
D. such that `(p,q) = (p, r) = (q, r) = 1`

Answer

To solve the question, we need to analyze the relationship between the Highest Common Factor (HCF) and the Least Common Multiple (LCM) of three natural numbers \( p, q, r \). The question states that the product of the HCF and LCM of these numbers equals the product of the numbers themselves, i.e.,

\[ \text{HCF}(p, q, r) \times \text{LCM}(p, q, r) = p \times q \times r \]

### Step-by-step Solution:

1. **Understanding HCF and LCM**:
- The HCF of a set of numbers is the largest number that divides all of them without leaving a remainder.
- The LCM of a set of numbers is the smallest number that is a multiple of all of them.

2. **Using the Property**:
- The property that states \( \text{HCF}(a, b) \times \text{LCM}(a, b) = a \times b \) can be extended to three numbers. Thus, for three numbers \( p, q, r \):
\[ \text{HCF}(p, q, r) \times \text{LCM}(p, q, r) = p \times q \times r \]

3. **Analyzing the Condition**:
- For the equation to hold true, the HCF must be a divisor of each of the numbers \( p, q, r \).
- If \( p, q, r \) are not prime numbers, they would have common factors, which would affect the HCF and LCM.

4. **Considering Prime Numbers**:
- If \( p, q, r \) are prime numbers, their HCF is 1 (since prime numbers have no common factors other than 1).
- The LCM of prime numbers \( p, q, r \) is simply their product \( p \times q \times r \) because they do not share any factors.

5. **Calculating HCF and LCM**:
- For prime numbers \( p, q, r \):
\[ \text{HCF}(p, q, r) = 1 \]
\[ \text{LCM}(p, q, r) = p \times q \times r \]
- Therefore, substituting back into the equation:
\[ 1 \times (p \times q \times r) = p \times q \times r \]
- This confirms that the condition holds true.

6. **Conclusion**:
- The only case where the equation \( \text{HCF}(p, q, r) \times \text{LCM}(p, q, r) = p \times q \times r \) holds true is when \( p, q, r \) are all prime numbers.

### Final Answer:
Thus, \( p, q, r \) must be prime numbers.

---