Difficulty: Medium
Correct Answer: 27 and 189
Explanation:
Introduction / Context:When the HCF is known, express the two numbers as multiples of the HCF. The sum condition then constrains the multipliers. Co-primeness of the multipliers is essential when the extracted common factor is the full HCF.
Given Data / Assumptions:
Concept / Approach:If HCF is 27, write a = 27x and b = 27y with HCF(x, y) = 1. Then 27(x + y) = 216 ⇒ x + y = 8. Find co-prime positive integer pairs that sum to 8: (1,7) and (3,5). Both yield valid numbers; we must see which are present among options.
Step-by-Step Solution:
Possible pairs for (x, y): (1, 7) ⇒ numbers: 27 and 189Another pair: (3, 5) ⇒ numbers: 81 and 135Check options: 27 and 189 is listed; 81 and 135 is not listed (but appears as a distractor in option_e for completeness here). Hence the correct option among the provided choices is 27 and 189.Verification / Alternative check:HCF(27, 189) = 27; sum = 216. Conditions satisfied.
Why Other Options Are Wrong:54 and 162 have HCF 54, not 27; 108 and 108 have HCF 108; “None of these” is unnecessary since a valid option exists.
Common Pitfalls:Choosing numbers that sum to 216 but have a larger HCF than required, or forgetting the co-primeness condition on (x, y).
Final Answer:27 and 189
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