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

Let's analyze both the assertion (A) and the reason (R) given in the question:

1. Assertion (A): \(H.C.F. (36 m^{2}, 18 m) = 18 m\), where \(m\) is a prime number.
   • To solve this, first express each term using their prime factors:
     • \(36 m^{2} = 2^2 \times 3^2 \times m^2\)
     • \(18 m = 2 \times 3^2 \times m\)
   • The H.C.F. (Highest Common Factor) is obtained by taking the minimum power of all prime factors available in both expressions:
     • For 2: The minimum power is \(2^1 = 2\)
     • For 3: The minimum power is \(3^2 = 9\)
     • For \(m\): The minimum power is \(m^1 = m\)
   • Therefore, the H.C.F. is \(2 \times 9 \times m = 18 m\), proving that the assertion (A) is true.
2. Reason (R): The H.C.F. of two numbers is always less than or equal to the smaller number.
   • This is a well-established mathematical fact. When you find the H.C.F. of two numbers, it should divide both numbers. Hence, it will always be less than or equal to the smaller of the two numbers.
   • Therefore, the reason (R) is true.

Now let's analyze if the reason provides a correct explanation for the assertion:

• The assertion (A) is about calculating the H.C.F. of two algebraic expressions and confirming it as \(18 m\).
• The reason (R) states a general property of H.C.F., not specifically related to the relationship in the assertion.
• Thus, although both statements are independently true, the reason does not explain why the assertion is true.

In conclusion, the correct option is: Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).