Question

Assertion (A) : For two prime numbers \(x\) and \(y\) (\(x<y\)), HCF\((x, y) = x\) and LCM\((x, y) = y\). Reason (R): HCF\((x, y) \leq \) LCM\((x, y)\), where \(x, y\) are any two natural numbers.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Hint:

For two prime numbers, HCF is 1 and LCM is their product.

Answer 1

The Correct Option is D

Given:
- Assertion (A): For two prime numbers \(x\) and \(y\) with \(x < y\), HCF\((x, y) = x\) and LCM\((x, y) = y\).
- Reason (R): HCF\((x, y) \leq \) LCM\((x, y)\), where \(x, y\) are any two natural numbers.

Step 1: Analyze Assertion (A)
- If \(x\) and \(y\) are distinct prime numbers, then HCF\((x, y) = 1\), not \(x\).
- LCM\((x, y) = x \times y\) since primes have no common factors except 1.
- Therefore, Assertion (A) is false.

Step 2: Analyze Reason (R)
- For any two natural numbers \(x, y\), HCF\((x, y) \leq\) LCM\((x, y)\) always holds true.
- So, Reason (R) is true.

Final Answer:
Assertion (A) is false, but Reason (R) is true.