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 the Assertion using the divisibility idea rather than full factorisation.
Since $36m^2 = 2m \times 18m$, it's clear $18m$ completely divides $36m^2$, and $18m$ is the largest such common factor, so H.C.F. $=18m$. Assertion (A) is true.
Step 2: Check the Reason on its own merit.
It is a standard fact that the H.C.F. of two numbers can never exceed the smaller one, since the H.C.F. must divide both numbers. So Reason (R) is also true.
Step 3: Test whether (R) actually explains (A).
Reason (R) only gives an upper bound ($\text{H.C.F.} \le$ smaller number); it never tells us why the H.C.F. equals exactly $18m$. The real reason is that $18m$ divides $36m^2$ completely.
Step 4: Conclude.
Both statements are true, but (R) does not explain (A), matching option (2).
\[ \boxed{\text{Option (2)}} \]