If the product of two positive integers is 4941 and their Least Common Multiple (LCM) is 81, find the Highest Common Factor (HCF) of the two numbers.

Introduction / Context:
This question again applies the relationship between the product of two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM). Such relationships are a standard part of quantitative aptitude syllabi. When the product and LCM are known, the HCF can be obtained directly using a simple division. Understanding this saves a lot of time when dealing with large numbers where manual factorization is not practical.

Given Data / Assumptions:

• Product of the two numbers = 4941.
• LCM of the two numbers = 81.
• The numbers are positive integers.
• We need to find the HCF of the two numbers.

Concept / Approach:
The standard formula for any two positive integers a and b is: a * b = HCF(a, b) * LCM(a, b). We can use this to find one unknown when the other three quantities are known. In this problem, we know the product and the LCM, and we want the HCF. Rearranging the formula, we get: HCF = (product of the numbers) / (LCM). This is a direct and efficient method especially when the product is exactly divisible by the LCM, as it should be for valid integers.

Step-by-Step Solution:
Step 1: Write the relationship formula. a * b = HCF * LCM Step 2: Substitute given values. 4941 = HCF * 81 Step 3: Solve for HCF. HCF = 4941 / 81 Step 4: Perform the division. 4941 / 81 = 61 Therefore, the HCF of the two numbers is 61.

Verification / Alternative check:
We can perform a quick verification by multiplying the obtained HCF and the given LCM: 61 * 81. Calculating this gives 61 * 81 = 4941, which matches the given product exactly. This confirms that the values are consistent with the fundamental formula. If the multiplication did not give the original product, it would mean either the division was done incorrectly or some misunderstanding of the relationship occurred.

Why Other Options Are Wrong:
60: 60 * 81 = 4860, which is less than 4941, so 60 cannot be the HCF in this situation.
59: 59 * 81 = 4779, which does not match the product 4941.
35: 35 * 81 = 2835, far from the required product, so this is incorrect.
27: 27 * 81 = 2187, again very different from the given product, so it cannot be right.

Common Pitfalls:
One common mistake is to forget the exact formula and mix up HCF and LCM or attempt to factorize the product into two numbers directly, which can be time consuming. Another pitfall is to perform the division incorrectly, especially when working without a calculator. It is helpful to check the final result by multiplying HCF and LCM back to see if the original product is recovered. This extra verification step is quick but very effective in avoiding silly errors in exams.

Final Answer:
The HCF of the two numbers is 61.