H.C.F–L.C.M relationship: The H.C.F of two numbers is 16 and their L.C.M is 160. If one number is 32, find the other number.

Introduction / Context:
For two integers a and b, the fundamental relation is a * b = HCF(a, b) * LCM(a, b). With one number known, you can immediately find the other by rearranging this relation.

Given Data / Assumptions:

• HCF = 16
• LCM = 160
• One number = 32

Concept / Approach:
Compute product a * b = 16 * 160 = 2560. If a = 32, then b = 2560 / 32. Ensure the result is an integer and matches the HCF/LCM structure (i.e., multiples align correctly).

Step-by-Step Solution:

Verification / Alternative check:
HCF(32, 80) = 16 and LCM(32, 80) = (32 * 80) / 16 = 160. Both conditions satisfy the given information.

Why Other Options Are Wrong:
48, 96, 112, or 64 do not simultaneously satisfy the HCF 16 and LCM 160 conditions with 32.

Common Pitfalls:
Forgetting the product relation or dividing the wrong way around. Always compute other = (HCF * LCM) / known.

Final Answer:
80