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

Introduction:
This problem checks how well you understand the relationship between the highest common factor (H.C.F.), least common multiple (L.C.M.), and the actual numbers. It also tests your ability to use factorisation and simple inequalities to select the correct number from a given range.

Given Data / Assumptions:

• H.C.F. of the two numbers = 11.
• L.C.M. of the two numbers = 385.
• One of the numbers lies between 75 and 125.
• We need to find that number.

Concept / Approach:
Let the two numbers be 11m and 11n, since their H.C.F. is 11. Then, if m and n are co-prime, the L.C.M. of the two numbers is: LCM = 11 * m * n We are given that LCM = 385, so we can solve for m * n. We then find all pairs of co-prime factors (m, n) of this product and check which resulting number lies between 75 and 125.

Step-by-Step Solution:
Step 1: Let the numbers be 11m and 11n with gcd(m, n) = 1. Step 2: LCM(11m, 11n) = 11 * m * n. Step 3: Given LCM = 385, so 11 * m * n = 385. Step 4: Therefore, m * n = 385 / 11 = 35. Step 5: Factor pairs of 35 are (1,35) and (5,7), both co-prime pairs. Step 6: Possible numbers are 11 * 1 = 11, 11 * 35 = 385, and 11 * 5 = 55, 11 * 7 = 77. Step 7: Among these, the numbers between 75 and 125 are only 77.

Verification / Alternative check:
Check the pair (55, 77): H.C.F.(55,77) = 11 and L.C.M.(55,77) = (55 * 77) / 11 = 385. This pair satisfies the given conditions, and one of the numbers, 77, falls in the required range. Therefore, 77 is the correct choice.

Why Other Options Are Wrong:
99: With 99 as one number, it is not possible to pair another integer so that H.C.F. = 11 and L.C.M. = 385. 110: This number exceeds the structure implied by the factors of 385 and 11. 88 and 121: Neither fits as a member of any pair with H.C.F. 11 and L.C.M. 385. Any number not derived from the factor pairs of 35 will violate the H.C.F. and L.C.M. conditions.

Common Pitfalls:
Students sometimes forget that after factoring out the H.C.F., the remaining co-factors must be co-prime. Others may guess numbers in the range without checking both H.C.F. and L.C.M. conditions, leading to incorrect answers.

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