Product and G.C.D (interpreted as H.C.F): The product of two two-digit numbers is 2160 and their greatest common divisor is 12. Identify the numbers.

Introduction / Context:
For any two positive integers a and b, the relation a * b = GCD(a, b) * LCM(a, b) holds. When product and GCD are given, you can confirm candidate pairs by checking both the product and the GCD requirement.

Given Data / Assumptions:

• a * b = 2160
• GCD(a, b) = 12
• a and b are two-digit numbers.

Concept / Approach:
Test options: both numbers must be multiples of 12 to have GCD 12 (not strictly necessary, but typical); more importantly, confirm GCD and product exactly. Dismissing pairs that fail the GCD check is fastest.

Step-by-Step Solution:

Verification / Alternative check:
Compute LCM using a * b / GCD = 2160 / 12 = 180. Indeed, LCM(36, 60) = 180, confirming consistency.

Why Other Options Are Wrong:
They fail either the stated GCD or the product condition. “None of these” is incorrect because a valid pair exists.

Common Pitfalls:
Assuming any factor pair of 2160 works without checking the GCD; or miscomputing the GCD due to incomplete factorization.

Final Answer:
36 and 60