The product of two positive integers is 4941 and their least common multiple (LCM) is 81. What is the highest common factor (HCF) of these two numbers?

Introduction:This question evaluates your grasp of the standard relationship between the highest common factor (HCF), least common multiple (LCM) and the product of two numbers. It is a direct formula-based problem that can be solved very quickly once you know the rule.

Given Data / Assumptions:

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

Concept / Approach:For any two positive integers a and b, the following well-known relationship holds:a * b = HCF(a, b) * LCM(a, b)Rearranging this formula allows us to compute the HCF when the product and LCM are known:HCF(a, b) = (a * b) / LCM(a, b)

Step-by-Step Solution:Step 1: Start from the formula.a * b = HCF * LCMStep 2: Substitute given values.4941 = HCF * 81Step 3: Solve for HCF.HCF = 4941 / 81Step 4: Do the division.81 * 60 = 4860, remainder = 8181 * 61 = 4860 + 81 = 4941Therefore, HCF = 61

Verification / Alternative check:Once we get HCF = 61, we can imagine two numbers whose product is 4941 and which share this factor. Writing one number as 61 * p and the other as 61 * q, their product becomes 61 * 61 * p * q = 4941. Dividing 4941 by 61 gives 81, so p * q = 81, and LCM(61p, 61q) must be 81 * 61 / 61 = 81, matching the given LCM.

Why Other Options Are Wrong:60: 60 * 81 = 4860, which is less than 4941.59: 59 * 81 = 4779, not equal to 4941.35: 35 * 81 = 2835, far from the required product.27: 27 * 81 = 2187, also incorrect.

Common Pitfalls:Typical mistakes include attempting to factorize 4941 manually without using the formula, or randomly trying divisors. Always remember that product = HCF * LCM simplifies such questions greatly.

Final Answer:The highest common factor (HCF) of the two numbers is 61.