The L.C.M. of two numbers is 2376 and their H.C.F. is 33. If one number is 297, find the other number.
Aptitude
Problems on H.C.F and L.C.M
Difficulty: Easy
Choose an option
-
A216
-
B264
-
C642
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D792
-
E198
Answer
Correct Answer: 264
Explanation
Introduction / Context:The fundamental identity for two positive integers a and b is a*b = gcd(a, b) * lcm(a, b). Knowing the LCM, HCF, and one number allows us to compute the other directly by rearrangement.
Given Data / Assumptions:
- LCM = 2376.
- HCF = 33.
- One number a = 297.
- Find the other number b.
Concept / Approach:Use b = (HCF * LCM) / a. Ensure a divides the product exactly, since the result must be an integer. Perform simplification before multiplication if convenient.
Step-by-Step Solution:
Compute b = (33 * 2376) / 297.Observe 2376 / 297 = 8 (since 297 * 8 = 2376).Therefore b = 33 * 8 = 264.Verification / Alternative check:Check gcd(297, 264) = 33 and lcm(297, 264) = (297*264)/33 = 2376. Both match the given data, confirming correctness.
Why Other Options Are Wrong:
- 216, 642, 792, 198: These do not satisfy both the specified HCF and LCM simultaneously with 297 under the product identity.
Common Pitfalls:
- Miscalculating the quotient 2376/297.
- Forgetting to divide by the known number after multiplying HCF and LCM.
Final Answer:264