Question

Assertion (A): H.C.F. (36 m², 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.

Options

• Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A).

• Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A).

• Assertion (A) is true, but Reason (R) is false.

• Assertion (A) is false, but Reason (R) is true.

Answer 1

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

Explanation:

Evaluating Assertion (A):

We need to find the Highest Common Factor (H.C.F.) of 36 m² and 18 m, where m is a prime number.

First, factorize both terms:

36 m² = 2 × 18 × m × m

= 2 × 18 m × m

18 m = 1 × 18 m

The highest common factor present in both terms is 18 m.

Since m is a prime number (m ≥ 2), 18 m is a non-zero integer.

Therefore, H.C.F. (36 m², 18 m) = 18 m is true.

Evaluating Reason (R):

The H.C.F. of any two positive integers a and b is a divisor of both.

By definition, a positive divisor of a number cannot be greater than the number itself.

Thus, H.C.F. (a, b) ≤ a and H.C.F. (a, b) ≤ b.

This means the H.C.F. is always less than or equal to the smaller of the two numbers.

Therefore, Reason (R) is true.