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: Check Assertion (A) using a division check instead of prime factorisation.
Divide $36m^2$ by $18m$: $36m^2 \div 18m = 2m$, with remainder $0$. Since $18m$ divides $36m^2$ exactly, and $18m$ obviously divides itself, $18m$ is a common factor; being the larger of the two terms' smaller one, it is the H.C.F. So Assertion (A) is true.
Step 2: Check Reason (R) using the definition of a divisor.
For any two numbers, their H.C.F. must divide the smaller one, and a divisor of a number can never exceed that number. So H.C.F. is always $\le$ the smaller number. Reason (R) is true in general.
Step 3: Check if (R) explains (A).
Reason (R) only gives an upper bound; it does not explain why the H.C.F. is exactly $18m$ (that comes from $18m$ being a complete divisor of $36m^2$, not merely from being small). So (R) does not explain (A).
\[ \boxed{\text{Both (A) and (R) are true, but (R) is not the correct explanation of (A).}} \]