Two numbers are in the ratio 5 : 6 and their least common multiple (LCM) is 480. Determine the highest common factor (HCF) of the two numbers, showing clear reasoning.

Introduction / Context:
This problem connects the ratio of two numbers with their least common multiple (LCM) to find the highest common factor (HCF). In number theory, the product of two numbers equals the product of their HCF and LCM. When a ratio is given, representing the numbers as multiples of a common factor streamlines the calculation.

Given Data / Assumptions:

• Ratio of numbers = 5 : 6
• LCM = 480
• Both numbers are positive integers

Concept / Approach:
If two numbers are 5k and 6k for some positive integer k, then HCF(5k, 6k) = k (since 5 and 6 are coprime) and LCM(5k, 6k) = 30k. We are given the LCM, so we can solve for k directly. The HCF is then k itself.

Step-by-Step Solution:
Let the numbers be 5k and 6k.LCM(5k, 6k) = 30k.Given LCM = 480 ⇒ 30k = 480.Solve for k: k = 480 / 30 = 16.Therefore, HCF = k = 16.

Verification / Alternative check:
Compute numbers explicitly: 5k = 80 and 6k = 96. Then HCF(80, 96) = 16 and LCM(80, 96) = (80*96)/16 = 480, confirming consistency.

Why Other Options Are Wrong:
20 and 24 arise from misusing the product-HCF-LCM relation; 6 and 5 incorrectly treat the ratio terms themselves as the HCF; they ignore scaling by k.

Common Pitfalls:
Confusing the ratio numbers with the actual numbers; forgetting that LCM scales linearly with k when the ratio terms are coprime; or mixing up HCF and LCM in the identity number1 * number2 = HCF * LCM.

Final Answer:
16