Two numbers with given sum and H.C.F: The sum of two numbers is 216 and their H.C.F is 27. Which pair are the numbers?

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:

• a + b = 216
• HCF(a, b) = 27

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:

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