The Highest Common Factor (HCF) and Least Common Multiple (LCM) of two numbers are 11 and 825 respectively. If one of the numbers is 275, find the other number.

Introduction / Context:
This question is another application of the key relationship between the product of two numbers, their Highest Common Factor (HCF), and their Least Common Multiple (LCM). Such problems appear frequently in competitive exams, and solving them efficiently is a great way to save time. Once you remember the core formula, the calculation is direct, even if the numbers involved seem large at first glance.

Given Data / Assumptions:

• HCF of the two numbers = 11.
• LCM of the two numbers = 825.
• One of the numbers = 275.
• We need to find the other number.
• The numbers are positive integers.

Concept / Approach:
The fundamental formula for any two positive integers a and b is: a * b = HCF(a, b) * LCM(a, b). Here, HCF and LCM are provided, and one of the numbers is also given. We can simply rearrange the formula to find the missing second number. Specifically: other number = (HCF * LCM) / known number. This avoids manual factorization and provides a neat direct route to the answer.

Step-by-Step Solution:
Step 1: Write the formula. a * b = HCF * LCM Step 2: Let the other number be x. 275 * x = 11 * 825 Step 3: Multiply HCF and LCM. 11 * 825 = 9075 Step 4: Solve for x by dividing. x = 9075 / 275 Step 5: Simplify the division. 9075 / 275 = 33 Therefore, the other number is 33.

Verification / Alternative check:
We can verify by checking the HCF and LCM of 275 and 33. First, note that 275 = 11 * 25 and 33 = 11 * 3. The common factor is 11, so HCF = 11. Next, multiply the numbers: 275 * 33 = 9075. Using the formula, LCM = (product) / HCF = 9075 / 11 = 825. This matches the given LCM, confirming that our calculation is correct and consistent with the original data.

Why Other Options Are Wrong:
53, 45, 43, and 25: None of these values, when multiplied by 275 and then divided by 11, produce an LCM of 825. For example, if we take 45, the product is 275 * 45 = 12375; dividing by 11 gives 1125, not 825. Similar checks show that other options do not satisfy the given conditions for both HCF and LCM simultaneously.

Common Pitfalls:
Some students guess the other number by trying random divisors or factors of the LCM rather than using the direct formula. Others may confuse HCF and LCM, or perform multiplication and division inaccurately with larger numbers. To avoid such mistakes, always write down the relationship clearly and ensure that you handle each arithmetic operation carefully. Quick verification with the original data is also a good habit in exam settings.

Final Answer:
The other number is 33.