Q 91.
TCS aptitude question, verified with a worked answer. Free to practise - no sign-up.
The HCF of 2472, 1284 and a third number ‘N’ is 12. If their LCM is 2^3 × 3^2 × 5 × 103 × 107, then the number ‘N’ is: Solution- 2472=83103 1284=43107 so n = 43x dividing by hcf 2472/12=2103 1284/12=107 lcm/12=235103107 x=lcm/firstsecond x=235103107/2103107=35=15 number is = 1512=180 ans. d) - A) 2^2 × 3^2 × 7 - B) 2^2 × 3^3 × 103 - C) 2^2 × 3 × 5 - D) None of these
Show answer & explanation
The number N must be a multiple of the HCF (12) and its prime factors must be present in the LCM. By analyzing the prime factorizations of 2472, 1284, and the LCM, the missing factors required to satisfy the LCM are 3 and 5, making N = 12 * 15 = 180, which is 2^2 * 3 * 5.
Step-by-step Derivation:
Step 1: Find prime factorization of the given numbers.
2472 = 2^3 * 3 * 103
1284 = 2^2 * 3 * 107
Step 2: Identify the HCF and LCM requirements.
HCF(2472, 1284, N) = 12 = 2^2 * 3
LCM(2472, 1284, N) = 2^3 * 3^2 * 5 * 103 * 107
Step 3: Analyze the prime factors of the LCM relative to the known numbers.
- Power of 2: LCM has 2^3. 2472 already provides 2^3. N must have at least 2^2 (from HCF).
- Power of 3: LCM has 3^2. 2472 and 1284 only provide 3^1. Therefore, N must provide 3^2.
- Power of 5: LCM has 5^1. Neither 2472 nor 1284 has a factor of 5. Therefore, N must provide 5^1.
- Power of 103: LCM has 103^1. 2472 provides this.
- Power of 107: LCM has 107^1. 1284 provides this.
Step 4: Construct N based on the HCF and the missing LCM factors.
N must be a multiple of 12 (2^2 * 3).
To satisfy the LCM, N must include 3^2 and 5^1.
Since HCF is 12, N cannot have 2^3 (otherwise HCF would be 24 if other numbers allowed, but here it's limited by 1284's 2^2).
Wait, let's re-evaluate: HCF(2^33103, 2^23107, N) = 2^2*3. This means N must be of the form 2^2 * 3^k * 5^m ...
From LCM, we need 3^2 and 5^1. Since 2472 and 1284 only have 3^1, N must have 3^2.
Thus, N = 2^2 * 3^2 * 5 = 4 * 9 * 5 = 180.
Step 5: Compare with options.
Option C is 2^2 * 3 * 5 = 60.
Wait, the provided solution in the prompt says 15 * 12 = 180.
180 = 2^2 * 3^2 * 5.
Looking at Option C: 2^2 * 3 * 5 = 60.
If N = 60, LCM(2472, 1284, 60) = LCM(2^33103, 2^23107, 2^235) = 2^3 * 3 * 5 * 103 * 107.
But the LCM given is 2^3 * 3^2 * 5 * 103 * 107.
This means N must have 3^2.
N = 2^2 * 3^2 * 5 = 180.
Checking the options again:
A: 2^2 * 3^2 * 7 (Wrong, contains 7)
B: 2^2 * 3^3 * 103 (Wrong, contains 103 and 3^3)
C: 2^2 * 3 * 5 = 60 (Wrong, missing one power of 3)
However, the prompt's provided solution concludes 'ans. d)' but the logic in the prompt's text says '15*12=180'. 180 is not Option A, B, or C. Therefore, the correct choice is Option D (None of these).