Question

Directions: In the Question, a statement of Assertion
(A) is followed by a statement of Reason (R).

Choose the correct option from the following:

Assertion (A): For two odd prime numbers \(x\) and \(y\), \((x + y)\), \(\text{LCM}(2x,\ 4y) = 4xy\)
Reason (R):} \(\text{LCM}(x, y)\) is a multiple of \(\text{HCF}(x, y)\)

• (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.

Answer 1

The Correct Option is A

Given:
Assertion (A): For two odd prime numbers \(x\) and \(y\), \((x + y)\), \(\text{LCM}(2x, 4y) = 4xy\)
Reason (R): \(\text{LCM}(x, y)\) is a multiple of \(\text{HCF}(x, y)\)

Step 1: Analyze the Assertion (A)
- Since \(x\) and \(y\) are odd primes, their sum \(x + y\) is even.
- For LCM calculation:
\[ \text{LCM}(2x, 4y) \]
- Prime factorization:
\(2x = 2 \times x\), \(4y = 2^2 \times y\)
- Since \(x\) and \(y\) are distinct primes, their prime factors do not overlap.
- LCM is the product of the highest powers of all primes involved:
\[ \text{LCM}(2x, 4y) = 2^2 \times x \times y = 4xy \]
- So, Assertion (A) is true.

Step 2: Analyze the Reason (R)
- For any two numbers \(a\) and \(b\),
\[ \text{LCM}(a, b) \times \text{HCF}(a, b) = a \times b \] - This implies that LCM is always a multiple of HCF.
- Hence, Reason (R) is true.

Step 3: Check if Reason (R) explains Assertion (A)
- The LCM calculation in Assertion (A) uses prime factorization and the property involving HCF.
- Reason (R) explains why LCM involves multiples of HCF.
- Therefore, Reason (R) is the correct explanation of Assertion (A).

Final Answer:
(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).