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, what is the other number?

Introduction:
This question again uses the fundamental relationship between HCF, LCM and the product of two numbers. Such questions are designed to check how quickly you can convert verbal data into a mathematical formula and solve it.

Given Data / Assumptions:

• HCF of the two numbers = 11.
• LCM of the two numbers = 825.
• One of the numbers = 275.
• Find the other number.

Concept / Approach:
Use the identity for two positive integers a and b:
a * b = HCF(a, b) * LCM(a, b)With HCF, LCM and one number known, the other number can be obtained directly by division.

Step-by-Step Solution:
Step 1: Apply the formula.a * b = HCF * LCMStep 2: Substitute values.275 * (other number) = 11 * 825Step 3: Calculate the right-hand side.11 * 825 = 9075Step 4: Solve for the other number.Other number = 9075 / 275Step 5: Perform the division.275 * 33 = 9075, so the other number is 33

Verification / Alternative check:
Check HCF and LCM for 33 and 275. Factorize: 33 = 3 * 11, 275 = 5^2 * 11. Common factor is 11, so HCF = 11. LCM is (33 * 275) / 11 = 9075 / 11 = 825. Both values match the given information, confirming the answer.

Why Other Options Are Wrong:
45: 275 * 45 = 12375, not equal to 11 * 825 = 9075.
53: 275 * 53 = 14575, also incorrect.
75: 275 * 75 = 20625, does not match required product.
99: 275 * 99 = 27225, far greater than 9075.

Common Pitfalls:
Some candidates attempt to guess factors of 825 or 275 and get confused. Always rely on the direct formula and perform clean arithmetic. Factorization is only needed for verification, not for the main computation here.

Final Answer:
The other number is 33.