In number theory, find the highest common factor (HCF) of the three numbers 4 × 27 × 3125, 8 × 9 × 25 × 7, and 16 × 81 × 5 × 11 × 49.

Introduction:
This aptitude question tests your understanding of how to find the highest common factor (HCF) of large numbers that are expressed as products of smaller factors. Such questions become very simple when you use prime factorisation instead of trying to divide directly.

Given Data / Assumptions:

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

Concept / Approach:
The HCF of a set of numbers is found by taking the common prime factors with their smallest powers. So we first express each number as a product of prime numbers. Then we identify which primes are common to all three numbers and select the minimum exponent of each common prime. Finally, we multiply these to get the HCF.

Step-by-Step Solution:
First number: 4 × 27 × 3125 4 = 2^2, 27 = 3^3, 3125 = 5^5, so first number = 2^2 × 3^3 × 5^5 Second number: 8 × 9 × 25 × 7 8 = 2^3, 9 = 3^2, 25 = 5^2, 7 = 7, so second number = 2^3 × 3^2 × 5^2 × 7 Third number: 16 × 81 × 5 × 11 × 49 16 = 2^4, 81 = 3^4, 5 = 5, 11 = 11, 49 = 7^2, so third number = 2^4 × 3^4 × 5 × 7^2 × 11 Common primes to all three: 2, 3, and 5 Smallest exponent of 2 is 2, of 3 is 2, of 5 is 1 Therefore, HCF = 2^2 × 3^2 × 5 = 4 × 9 × 5 = 180

Verification / Alternative check:
We can check quickly: 180 divides each number because each number contains at least 2^2, 3^2, and 5 as factors. Also, no larger number with these primes will divide all three, since at least one number would not contain the higher power. So 180 is confirmed as the highest common factor.

Why Other Options Are Wrong:
360 is 2^3 × 3^2 × 5 and requires 2^3 in every number, which the first number does not have. 540 includes 3^3, which is not present in all three. 1260 contains extra prime factors (7) or higher powers, so it cannot be a common factor of all three numbers.

Common Pitfalls:
A common mistake is to multiply all visible numbers directly and then guess the HCF without doing factorisation. Another error is to include primes like 7 or 11 in the HCF just because they appear in one or two numbers rather than all three. Always work with prime factors systematically.

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