Question

Assertion (A) : H.C.F. ($36\text{ m}^2, 18\text{ m}$) = $18\text{ m}$, 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:

An H.C.F. of two terms $A$ and $B$ is exactly equal to $A$ (the smaller term) if and only if $A$ completely divides $B$.
This divisibility rule is the true reason for the Assertion, making Reason (R) merely a true general property, not the explanation.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given an Assertion (A) regarding the Highest Common Factor (H.C.F.) of two algebraic terms containing a prime variable $m$, and a Reason (R) which states a general inequality property of H.C.F.
We need to verify both statements and determine if the Reason explains the Assertion.

Step 2: Key Formula or Approach:
- The H.C.F. of two or more algebraic terms is the product of the lowest powers of the common factors.
- For any two positive integers $a$ and $b$, their H.C.F. is a divisor of both. Since any positive divisor of a number is less than or equal to that number, $\text{H.C.F.}(a, b) \le \min(a, b)$.

Step 3: Detailed Explanation:

• Evaluate Assertion (A):
Let us find the H.C.F. of the two terms $36m^2$ and $18m$:
- Factorizing the terms:
\[ 36m^2 = 2^2 \times 3^2 \times m^2 \]
\[ 18m = 2 \times 3^2 \times m \]
- Taking the lowest powers of the common bases:
\[ \text{H.C.F.} = 2^1 \times 3^2 \times m^1 = 18m \]
Since $m$ is a prime number (implying $m \ge 2$), these expressions are well-defined. Thus, the H.C.F. is indeed $18m$.
Therefore, Assertion (A) is true.

• Evaluate Reason (R):
The statement says: "H.C.F. of two numbers is always less than or equal to the smaller number."
Let the two numbers be $x$ and $y$ with $x \le y$.
Since the H.C.F. must divide $x$, it cannot be greater than $x$. Therefore, $\text{H.C.F.}(x, y) \le x$.
This is a fundamental and universally true mathematical property of H.C.F.
Therefore, Reason (R) is true.

• Determine if Reason (R) is the correct explanation of Assertion (A):
Although the H.C.F. $18m$ is equal to the smaller number (since $m$ is prime, $18m \lt 36m^2$), the reason why the H.C.F. is exactly $18m$ is because $18m$ is a complete divisor of $36m^2$.
The inequality in Reason (R) only states an upper bound constraint; it does not explain how the H.C.F. value of $18m$ was calculated.
Therefore, Reason (R) is not the correct explanation of Assertion (A).


Step 4: Final Answer:
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).