The LCM and HCF of two numbers are 4284 and 32 respectively. If one of the numbers is 672, what is the other number?

Introduction:
This question uses a key relationship between two numbers, their highest common factor (HCF), and their least common multiple (LCM). Given the LCM, HCF, and one number, you are asked to find the second number.

Given Data / Assumptions:

• LCM of two numbers = 4284
• HCF of the two numbers = 32
• One number = 672
• We must find the other number.

Concept / Approach:
For two positive integers a and b, there is a direct relation: a × b = HCF(a, b) × LCM(a, b). This formula allows us to compute the missing number when the other three values are known. We simply rearrange it to solve for the unknown number.

Step-by-Step Solution:
Let the unknown number be x. Given: HCF = 32, LCM = 4284, known number = 672. Use the relation: a × b = HCF × LCM So 672 × x = 32 × 4284 Compute the right side: 32 × 4284 = 137088 So 672 × x = 137088 Now solve for x: x = 137088 ÷ 672 137088 ÷ 672 = 204 Therefore, the other number is 204

Verification / Alternative check:
We can check that HCF(672, 204) is 32 and LCM(672, 204) is 4284. Using prime factorisation: 672 = 2^5 × 3 × 7, 204 = 2^2 × 3 × 17. The HCF is 2^2 × 3 = 32, and the LCM is the product of HCF and co-prime parts, which indeed yields 4284. This confirms that 204 is correct.

Why Other Options Are Wrong:
The options 64, 92, and 102, when multiplied with 672, do not satisfy the product equation HCF × LCM. They either give the wrong HCF, wrong LCM, or both. Therefore they cannot be the other number in this pair.

Common Pitfalls:
Students sometimes forget the formula relating HCF, LCM, and the product of two numbers, and instead try to guess factors. Others may make arithmetic mistakes while multiplying or dividing large numbers. Writing the product relation clearly and simplifying step by step reduces these errors.

Final Answer:
The second number whose HCF and LCM with 672 are 32 and 4284 is 204.