The HCF of 2472, 1284, and a third number 'n' is 12. If their LCM is 8 × 9 × 5…

Concept: For three numbers, the HCF is built from the LOWEST power of every prime common to all three, while the LCM is built from the HIGHEST power of every prime appearing in any of them. So when two of the three numbers are known along with the overall HCF and LCM, the third number's own prime factorisation must supply exactly the extra prime power(s) needed to reach the stated LCM, without disturbing the stated HCF.

Working:

1. Factorise the two given numbers: 2472 = 23 × 3 × 103 and 1284 = 22 × 3 × 107.

2. The third number must not introduce any prime absent from the given LCM (23 × 32 × 51 × 1031 × 1071), and it must supply whatever prime power the other two numbers don't already reach — here that means at least one more factor of 3 (to reach 32 from the existing 31) and a full factor of 5 (absent from both 2472 and 1284); it must also be a multiple of the stated HCF, 12 = 22 × 3.

3. Checking the given choices against these two conditions: a value with a factor of 71 introduces a prime the given LCM doesn't have, and a value built only from 2, 3, and 103 supplies no factor of 5 at all — both fail.

4. Only 22 × 32 × 51 = 4 × 9 × 5 = 180 supplies the missing 3 and 5, stays within the LCM's 23 ceiling, and is a multiple of 12 — so it is the value consistent with both stated conditions.

Cross-check: GCD(2472, 1284, 180) = 22 × 3 = 12, matching the given HCF; LCM(2472, 1284, 180) = 23 × 32 × 5 × 103 × 107, matching the given LCM exactly. (Other values — such as 23 × 32 × 51, or ones that also include 103 or 107 — would satisfy the two stated conditions too, but 180 is the one appearing among the given choices.)