Find the H.C.F. (greatest common divisor) of the following three numbers:\n4 × 27 × 3125, 8 × 9 × 25 × 7, and 16 × 81 × 5 × 11 × 49.

Introduction:
This question tests your ability to find the highest common factor (H.C.F.) of large numbers given in factorised product form. It reinforces prime factorisation skills and understanding of how to extract common factors across multiple composite numbers.

Given Data / Assumptions:
We are given three numbers in product form:

• First number: 4 × 27 × 3125
• Second number: 8 × 9 × 25 × 7
• Third number: 16 × 81 × 5 × 11 × 49
• We need to find the H.C.F. of these three numbers.

Concept / Approach:
To find the H.C.F. of several numbers, we:

• Express each number as a product of prime powers.
• For each prime, take the smallest exponent that appears in all the numbers.
• Multiply these common prime factors together to get the H.C.F.

Step-by-Step Solution:
Step 1: Factorise each component. 4 = 2^2, 27 = 3^3, 3125 = 5^5. First number = 2^2 * 3^3 * 5^5. 8 = 2^3, 9 = 3^2, 25 = 5^2, 7 = 7^1. Second number = 2^3 * 3^2 * 5^2 * 7^1. 16 = 2^4, 81 = 3^4, 5 = 5^1, 11 = 11^1, 49 = 7^2. Third number = 2^4 * 3^4 * 5^1 * 7^2 * 11^1. Step 2: Identify common primes: 2, 3, and 5 appear in all three numbers. Step 3: Take minimum exponents across the three numbers. For 2: min(2, 3, 4) = 2. For 3: min(3, 2, 4) = 2. For 5: min(5, 2, 1) = 1. Step 4: H.C.F. = 2^2 * 3^2 * 5^1 = 4 * 9 * 5 = 180.

Verification / Alternative check:
We can divide all three original numbers by 180 and confirm that the quotients are integers, and that no higher common factor exists. Since we already used minimum prime exponents, 180 is guaranteed to be the greatest common divisor.

Why Other Options Are Wrong:
360 and 540: These include higher powers of 2 or 3 than are common to all three numbers. 1260: This introduces extra factors like 7 and 11 that are not common to all three numbers. 90: This is a common factor but not the greatest, since 180 is larger and still divides all three numbers.

Common Pitfalls:
Errors often arise from miscalculating prime powers, forgetting that H.C.F. uses minimum exponents, or mixing H.C.F. with L.C.M. (which uses maximum exponents). Careful prime factorisation avoids these issues.

Final Answer:
The highest common factor of the three given numbers is 180.