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 contribute to the overall LCM without exceeding the powers present in the LCM. Based on the prime factorization of the given numbers and the LCM, N must be 12 * 15 = 180, which is 2^2 * 3 * 5.
Step-by-step Derivation:
Step 1: Prime factorization of the given numbers:
2472 = 2^3 * 3 * 103
1284 = 2^2 * 3 * 107
Step 2: Analyze the HCF (Highest Common Factor):
HCF(2472, 1284, N) = 12 = 2^2 * 3.
This implies N must be a multiple of 12, so N = 2^2 * 3 * k, where k is an integer.
Step 3: Analyze the LCM (Least Common Multiple):
LCM = 2^3 * 3^2 * 5 * 103 * 107.
Comparing the LCM to the prime factors of 2472 and 1284:
- Power of 2: Max(3, 2, power in N) = 3. (Satisfied by 2472)
- Power of 3: Max(1, 1, power in N) = 2. Therefore, N must contain 3^2.
- Power of 5: Max(0, 0, power in N) = 1. Therefore, N must contain 5^1.
- Power of 103: Max(1, 0, power in N) = 1. (Satisfied by 2472)
- Power of 107: Max(0, 1, power in N) = 1. (Satisfied by 1284)
Step 4: Determine N:
From the LCM analysis, N must provide the missing factors 3^1 (to make 3^2) and 5^1.
Since N must be a multiple of the HCF (2^2 * 3), we have:
N = (2^2 * 3) * (3 * 5) = 2^2 * 3^2 * 5 = 180.
Step 5: Evaluate the options:
Option A: 2^2 * 3^2 * 7 (Incorrect, contains 7)
Option B: 2^2 * 3^3 * 103 (Incorrect, power of 3 is too high)
Option C: 2^2 * 3 * 5 = 60. (Wait, let's re-verify the provided solution in the prompt).
Re-evaluating the prompt's provided solution: The prompt's solution says N = 180. 180 = 2^2 * 3^2 * 5.
Looking at Option C: 2^2 * 3 * 5 = 60.
If N = 60, HCF(2472, 1284, 60) = 12. LCM(2472, 1284, 60) = LCM(2^33103, 2^23107, 2^235) = 2^3 * 3 * 5 * 103 * 107.
But the given LCM is 2^3 * 3^2 * 5 * 103 * 107.
Therefore, N must have 3^2. N = 2^2 * 3^2 * 5 = 180.
Since 180 is not Option A, B, or C (Option C is 60), the correct answer should be Option D (None of these). However, the prompt's internal solution text claims 'ans. d)' but then lists the calculation resulting in 180. Let's check Option C again: 2^2 * 3 * 5 = 60. 180 is 2^2 * 3^2 * 5.
Correcting the logic: The prompt's solution says 'ans. d)'. Option D is 'None of these'. 180 is not listed in A, B, or C. Thus, Option D is the correct choice.