The highest common factor (HCF) of two numbers is 11 and their least common multiple (LCM) is 330. If one of the numbers is 55, what is the other number?

Introduction:
This problem checks your understanding of the relationship between the highest common factor (HCF), the least common multiple (LCM), and the product of two numbers. It is a very common type of question in aptitude and bank exams.

Given Data / Assumptions:

• HCF of the two numbers = 11.
• LCM of the two numbers = 330.
• One of the numbers = 55.
• We must find the other number.

Concept / Approach:
For any two positive integers a and b, the fundamental relationship is:
a * b = HCF(a, b) * LCM(a, b)This allows us to find the missing number when we know the HCF, LCM, and one of the numbers.

Step-by-Step Solution:
Step 1: Use the formula.a * b = HCF * LCMStep 2: Substitute the known values.55 * (other number) = 11 * 330Step 3: Compute the right-hand side first.11 * 330 = 3630Step 4: Solve for the other number.Other number = 3630 / 55Step 5: Perform the division.55 * 66 = 3630, so the other number is 66

Verification / Alternative check:
Let the two numbers be 55 and 66. Their HCF is 11 (since 55 = 5 * 11 and 66 = 6 * 11), and their LCM is (55 * 66) / 11 = 3630 / 11 = 330. This matches the given HCF and LCM, confirming our answer.

Why Other Options Are Wrong:
33: The product 55 * 33 = 1815, which does not equal 11 * 330 = 3630.
44: 55 * 44 = 2420, again not equal to 3630.
77: 55 * 77 = 4235, does not satisfy the relation.
99: 55 * 99 = 5445, also incorrect.

Common Pitfalls:
Some learners try to guess the other number or confuse HCF and LCM. Others forget the crucial formula connecting product, HCF and LCM. Always start with the formula to avoid mistakes.

Final Answer:
The other number is 66.