Recover two numbers from LCM, HCF, and ratio The LCM and HCF of two numbers are 240 and 16, respectively. If their ratio is 3 : 5, find the two numbers.

Introduction / Context:Two unknown numbers are given indirectly via their LCM, HCF, and ratio. When the ratio is a : b with gcd(a, b) = 1, the actual numbers are a * HCF and b * HCF. We can verify the LCM at the end.

Given Data / Assumptions:

• LCM = 240.
• HCF = 16.
• Ratio = 3 : 5 (coprime pair).

Concept / Approach:Let numbers be 3x and 5x. Their gcd equals x because 3 and 5 are coprime. Given HCF = 16 ⇒ x = 16, so numbers are 48 and 80. Verify LCM using the identity LCM * HCF = product.

Step-by-Step Solution:Assume numbers = 3x and 5x; gcd(3x, 5x) = x.Given HCF = 16 ⇒ x = 16.Numbers: 48 and 80.Check LCM: (48 * 80) / gcd(48, 80) = (3840) / 16 = 240, matches.

Verification / Alternative check:Prime factors: 48 = 2^4 * 3; 80 = 2^4 * 5. LCM takes 2^4 * 3 * 5 = 240; gcd takes 2^4 = 16.

Why Other Options Are Wrong:Other pairs either do not have gcd 16 or their LCM is not 240 when checked.

Common Pitfalls:Forgetting that the given ratio terms must be coprime to apply the simple multiplication by HCF, or miscomputing the LCM check.

Final Answer:48, 80