The H.C.F. and L.C.M. of two numbers are 11 and 385 respectively. If one of the numbers lies between 75 and 125, what is that number?

Introduction / Context:
This question requires you to use the relation between H.C.F., L.C.M., and the original numbers to find specific values under a given range constraint. Such problems are standard in aptitude tests to evaluate your ability to connect number theoretic properties with inequality conditions.

Given Data / Assumptions:

• H.C.F. of the two numbers is 11.
• L.C.M. of the two numbers is 385.
• One of the numbers lies between 75 and 125.
• We are asked to identify that number.

Concept / Approach:
Let the two numbers be 11a and 11b, where a and b are co-prime integers. Then their L.C.M. is 11ab. We are told the L.C.M. is 385, so 11ab = 385. From this we find ab. The possible co-prime pairs (a, b) that satisfy this equation lead to the actual numbers. Finally, we check which of these numbers lies within the given range 75 to 125.

Step-by-Step Solution:
Step 1: Let the numbers be 11a and 11b with H.C.F.(a, b) = 1.Step 2: L.C.M.(11a, 11b) = 11ab.Step 3: Given that L.C.M. = 385, so 11ab = 385.Step 4: Divide both sides by 11: ab = 385 / 11 = 35.Step 5: The factor pairs of 35 are (1, 35) and (5, 7), and their reverses.Step 6: Both pairs (1, 35) and (5, 7) have co-prime components.Step 7: Corresponding numbers are: For (1, 35) we get 11 and 385. For (5, 7) we get 55 and 77.Step 8: Among 11, 385, 55, and 77, the numbers lying between 75 and 125 are 77 only.

Verification / Alternative check:
Check 55 and 77 as a possible pair: H.C.F.(55, 77) = 11 and L.C.M. = (55 * 77) / 11 = 385. This confirms that 55 and 77 are valid numbers for the given H.C.F. and L.C.M. Only 77 lies in the required range, so it must be the answer.

Why Other Options Are Wrong:
Option b (88), option c (99), and option d (110) do not pair with any integer in such a way that H.C.F. becomes 11 and L.C.M. remains 385. They either produce a different product or lead to the wrong L.C.M. when combined with any compatible pair. A quick check using the formula L.C.M. = (product) / H.C.F. shows that they do not yield 385.

Common Pitfalls:
Students sometimes attempt trial and error directly on answer choices, which can be time-consuming and error-prone. Others forget that the numbers must be multiples of 11 because the H.C.F. is 11. Working with the simplified variables a and b and using ab = 35 provides a structured path to the solution.

Final Answer:
The required number between 75 and 125 is 77.