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, find the other number using the relationship between HCF, LCM, and the product of the numbers.

Introduction / Context:
This question checks whether you know how to connect HCF, LCM, and the product of two numbers. Such problems are standard in number system topics in aptitude exams. By using a single formula, we can move quickly from given values to the unknown number without trial and error. Understanding this relationship is not only useful for direct HCF and LCM questions but also for many mixture and ratio problems where similar reasoning appears in disguised forms.

Given Data / Assumptions:

• HCF of two numbers = 11.
• LCM of the two numbers = 330.
• One of the numbers = 55.
• Both numbers are positive integers.
• We need to find the other number.

Concept / Approach:
For any two positive integers a and b, the relationship between HCF and LCM is: a * b = HCF(a, b) * LCM(a, b). If we know one number, the HCF, and the LCM, we can rearrange this formula to find the second number. Specifically, other number = (HCF * LCM) / known number. This formula is very efficient and avoids the need to factorize the numbers manually.

Step-by-Step Solution:
Step 1: Write the standard formula. a * b = HCF * LCM Step 2: Substitute the known values into the formula. Let the other number be x. 55 * x = 11 * 330 Step 3: Calculate the right hand side. 11 * 330 = 3630 Step 4: Solve for x. x = 3630 / 55 Step 5: Perform the division. 3630 / 55 = 66 Therefore, the other number is 66.

Verification / Alternative check:
We can verify the answer by checking both the HCF and LCM. The two numbers are 55 and 66. Their HCF is 11 because both are divisible by 11 and no higher common factor exists. To check the LCM, we can use the formula again: LCM = (55 * 66) / 11 = 3630 / 11 = 330. This matches the given LCM, so the calculation is consistent. Hence the result is correct.

Why Other Options Are Wrong:
33: If the other number were 33, then the product would be 55 * 33 = 1815, leading to an incorrect LCM when combined with HCF 11.
44: With 44, the product is 55 * 44 = 2420; dividing by 11 gives LCM = 220, not 330.
77: With 77, product is 55 * 77 = 4235; dividing by 11 gives LCM = 385, again not equal to 330.
110: Here, product 55 * 110 = 6050, and 6050 / 11 = 550, which does not match the given LCM.

Common Pitfalls:
A common error is to try to guess the other number by random factorization or by relying only on HCF or only on LCM instead of using both together. Another mistake is to misapply the formula and write HCF * LCM = HCF + LCM or some other incorrect combination. Some students also forget to simplify the division correctly, leading to calculation mistakes. Always write the formula clearly, substitute carefully, and compute step by step.

Final Answer:
The other number is 66.