The highest common factor (HCF) of 3240, 3600, and a third number is 36, and their least common multiple (LCM) is 2^4 × 3^5 × 5^2 × 7^2. What is the value of this third number?

Introduction:
This is a challenging number theory problem involving both the highest common factor (HCF) and least common multiple (LCM) of three numbers. You must use prime factorisation and the relationships between HCF, LCM, and the given numbers to identify the unknown third number.

Given Data / Assumptions:

• First number = 3240
• Second number = 3600
• Third number = unknown
• HCF of all three numbers = 36
• LCM of all three numbers = 2^4 × 3^5 × 5^2 × 7^2

Concept / Approach:
For any set of numbers, the HCF is obtained by taking common primes with minimum exponents, while the LCM is obtained by taking all primes that appear with maximum exponents. We factorise the known numbers, compare them with the given LCM and HCF, and deduce the prime factorisation of the unknown third number so that all conditions are satisfied.

Step-by-Step Solution:
First factorise the known numbers: 3240 = 2^3 × 3^4 × 5 3600 = 2^4 × 3^2 × 5^2 Given LCM of the three numbers = 2^4 × 3^5 × 5^2 × 7^2 Given HCF of the three numbers = 36 = 2^2 × 3^2 Notice that 7 and 7^2 appear only in the LCM, so they must come entirely from the third number. Also, the LCM has 3^5, while current maximum from 3240 and 3600 is 3^4, so the third number must contribute at least 3^5. Because the HCF is 2^2 × 3^2, the third number must contain 2^2 and at least 3^2, but it must not add any factor of 5 beyond the given minimum pattern. From the HCF condition: gcd(360, third number) = 36. This forces the exponent of 2 in the third number to be exactly 2 and the exponent of 5 to be 0. Therefore the third number must be: 2^2 × 3^5 × 7^2 Compute: 2^2 = 4, 3^5 = 243, 7^2 = 49 Third number = 4 × 243 × 49 = 47628

Verification / Alternative check:
If we compute gcd(3240, 3600, 47628), we get 36. If we compute lcm(3240, 3600, 47628), we obtain 2^4 × 3^5 × 5^2 × 7^2, exactly as given. This double check confirms that 47628 satisfies both HCF and LCM conditions.

Why Other Options Are Wrong:
The other options do not produce the required prime factorisation conditions. For example, they either include extra prime factors of 5, miss the factor 7^2, or do not respect the HCF constraint of 36. None of them give the correct combination of HCF and LCM simultaneously.

Common Pitfalls:
Students often try to solve such questions using only approximate divisibility without doing complete prime factorisation. Another mistake is to ignore the HCF condition after matching the LCM, or vice versa. Both constraints must be checked carefully because they strongly restrict the possible third number.

Final Answer:
The required third number that satisfies the given HCF and LCM conditions is 47628.