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 Euclidean algorithm instead of prime factors.
Dividing $36m^2$ by $18m$ gives $36m^2 = 2m \times 18m$ with remainder $0$. Since the division leaves no remainder, $18m$ already divides $36m^2$ exactly, so $\text{H.C.F.}(36m^2, 18m) = 18m$. Assertion (A) is true.
Step 2: Check the Reason using a general rule.
For any two positive quantities $x \le y$, the H.C.F. must divide $x$, so it can never exceed $x$; this means $\text{H.C.F.}(x,y) \le x$ always. So Reason (R) is a true general statement.
Step 3: Decide whether R actually explains A.
Reason (R) only gives an upper limit on the H.C.F.; it does not tell us why the H.C.F. turns out to be exactly $18m$ rather than some smaller common factor. The real reason is the exact divisibility shown in Step 1, so R does not explain A.
Step 4: State the final match.
Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A).
\[ \boxed{\text{Both A and R are true, but R is not the correct explanation of A.}} \]