The least common multiple (LCM) and highest common factor (HCF) of two numbers are 4284 and 32 respectively. If one of the numbers is 672, find the other number using the relationship between LCM, HCF, and the product of two numbers.

Introduction / Context:
This question tests a standard and very useful identity connecting two numbers, their highest common factor (HCF), and their least common multiple (LCM). The product of two positive integers equals the product of their HCF and LCM. Using this identity, we can find an unknown number when the other number, the HCF, and the LCM are known.

Given Data / Assumptions:

• LCM of the two numbers = 4284
• HCF of the two numbers = 32
• One number = 672
• Both numbers are positive integers.

Concept / Approach:
For any two positive integers a and b, the fundamental relationship is: a * b = HCF(a, b) * LCM(a, b). If one number, the HCF, and the LCM are known, we can rearrange this to find the other number. Specifically, if a is known and b is unknown, then: b = (HCF * LCM) / a.

Step-by-Step Solution:
Let the unknown number be x. Given: LCM = 4284, HCF = 32, known number = 672. Use the identity: 672 * x = 32 * 4284. Compute the right side: 32 * 4284 = 137088. So 672 * x = 137088. Solve for x: x = 137088 / 672. Perform division: 137088 / 672 = 204. Therefore, the second number is 204.

Verification / Alternative check:
We can verify by recomputing HCF and LCM of 672 and 204. The HCF of 672 and 204 is 32, and the LCM is 4284. Also, 672 * 204 = 137088, which equals 32 * 4284, confirming that the identity holds and that the calculated number is correct.

Why Other Options Are Wrong:
102, 64, 92, and 156 do not satisfy the identity when paired with 672. For each of these, 672 multiplied by the option either does not equal 32 multiplied by 4284 or leads to inconsistent HCF and LCM values.

Common Pitfalls:
A common error is to add or subtract the LCM and HCF instead of using the product identity. Another mistake is to perform the division incorrectly or to forget that the HCF and LCM relationship applies only to pairs of numbers. Carefully applying the formula a * b = HCF * LCM prevents such mistakes.

Final Answer:
204