The highest common factor (HCF) of the numbers 3240, 3600, and a third unknown number is 36, and the least common multiple (LCM) of all three is 2^4 * 3^5 * 5^2 * 7^2. Using the relationship between HCF, LCM, and prime factorization, determine the value of the third number.

Introduction / Context:
This question tests a deeper understanding of how the highest common factor (HCF) and least common multiple (LCM) relate to the prime factorizations of three numbers. Instead of only computing HCF or LCM, we are asked to reconstruct an unknown number from the given combined HCF and LCM information, which is a more advanced application frequently seen in aptitude exams.

Given Data / Assumptions:

• HCF of 3240, 3600, and the third number = 36
• LCM of 3240, 3600, and the third number = 2^4 * 3^5 * 5^2 * 7^2
• First known number = 3240
• Second known number = 3600
• Third number is positive and must be consistent with the given HCF and LCM.

Concept / Approach:
For each prime, the exponent in the HCF is the minimum exponent among the three numbers, and the exponent in the LCM is the maximum exponent among them. By factoring the known numbers and comparing with the required HCF and LCM prime powers, we can deduce the prime factorization of the unknown third number and then compute it directly.

Step-by-Step Solution:
3240 = 2^3 * 3^4 * 5^1 3600 = 2^4 * 3^2 * 5^2 HCF = 36 = 2^2 * 3^2 LCM = 2^4 * 3^5 * 5^2 * 7^2 Let the third number be N = 2^a * 3^b * 5^c * 7^d. From HCF for prime 2: min(3, 4, a) = 2 so a = 2. From HCF for prime 3: min(4, 2, b) = 2 so b ≥ 2. From HCF for prime 5: min(1, 2, c) = 0 so c = 0. From LCM for prime 3: max(4, 2, b) = 5 so b = 5. From LCM for prime 7: max(0, 0, d) = 2 so d = 2. Thus N = 2^2 * 3^5 * 7^2 = 4 * 243 * 49 = 47628.

Verification / Alternative check:
We can check that N shares HCF 36 with the other numbers and that the LCM using all three numbers matches the given expression. The exponents for 2 and 3 satisfy both minimum and maximum constraints, while 5 and 7 appear with required exponents in the overall LCM. This confirms that 47628 is consistent with the given information.

Why Other Options Are Wrong:
35280 and 42146 either introduce incorrect prime factors or fail to match the HCF condition. 49874 and 24157 do not generate the specified LCM when combined with 3240 and 3600.

Common Pitfalls:
A frequent mistake is to assume that the product of numbers is always simply HCF multiplied by LCM, which is only guaranteed for two numbers, not for three. Another common error is ignoring the prime exponent rules for minimum (HCF) and maximum (LCM). Working prime by prime prevents such confusion.

Final Answer:
47628