Let the highest common factor (HCF) of two positive integers a and b be 12, with a > b > 12. Which ordered pair (a, b) satisfies these conditions?
Aptitude
Problems on H.C.F and L.C.M
Difficulty: Easy
Choose an option
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A12, 24
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B24, 12
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C24, 36
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D36, 24
Answer
Correct Answer: 36, 24
Explanation
Introduction / Context:This item checks understanding of the definition of HCF and the implications for the possible forms of a and b. If HCF(a, b) = 12, both a and b must be multiples of 12, and no larger common factor may divide both simultaneously.
Given Data / Assumptions:
- a and b are positive integers.
- HCF(a, b) = 12.
- a > b > 12.
Concept / Approach:Write a = 12x and b = 12y with HCF(x, y) = 1. The inequality a > b > 12 implies x > y ≥ 2 (since b > 12 must be at least 24). Check each option against these conditions and verify the HCF directly if needed.
Step-by-Step Solution:
Option D: a = 36, b = 24 → both multiples of 12.Compute HCF(36, 24): 36 = 12*3, 24 = 12*2, HCF = 12.The inequality holds: 36 > 24 > 12.Verification / Alternative check:
Other options fail one or more conditions: (12, 24) or (24, 12) violate a > b > 12; (24, 36) reverses order and b is not > 12 for the larger-first order; hence only (36, 24) works.Why Other Options Are Wrong:
- (12, 24) and (24, 12) do not satisfy a > b > 12 as required.
- (24, 36) does not have a > b; it is reversed and also b = 36 is not less than a.
Common Pitfalls:
- Forgetting that HCF imposes co-primeness on the scaled-down pair (x, y) and missing the strict order a > b > 12.
Final Answer:
36, 24