Question

Assertion (A) : H.C.F. \((36 m^{2}, 18 m) = 18 m\), where \(m\) is a prime number.
Reason (R) : H.C.F. of two numbers is always less than or equal to the smaller number.

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

Hint:

For any two numbers \(x\) and \(y\), if \(x\) divides \(y\), then \(HCF(x, y) = x\) and \(LCM(x, y) = y\). This is the specific logic often tested in such questions.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
Highest Common Factor (H.C.F.) is the largest positive integer that divides each of the integers. For monomials, H.C.F. is the product of the H.C.F. of coefficients and the lowest power of each common variable.
Step 2: Detailed Explanation:
Evaluating Assertion (A):
The expressions are \(36m^2\) and \(18m\).
Since \(m\) is a prime number, \(m \ge 2\). Thus \(36m^2\) is a multiple of \(18m\).
\[ 36m^2 = 18m \times 2m \]
If one number is a factor of another, the smaller number is the H.C.F.
So, \(H.C.F.(36m^2, 18m) = 18m\).
Assertion (A) is True.
Evaluating Reason (R):
By definition, H.C.F. of two positive integers \(a\) and \(b\) is always \(\le a\) and \(\le b\). Therefore, it is always less than or equal to the smaller of the two numbers.
Reason (R) is True.
Relationship Analysis:
While both statements are true, Reason (R) is a general property of H.C.F. and does not specifically explain why the H.C.F. of \(36m^2\) and \(18m\) is exactly \(18m\). The correct explanation for (A) would be that \(18m\) is a factor of \(36m^2\).
Step 3: Final Answer:
Both (A) and (R) are true but (R) is not the correct explanation.