Match List I with List II. Choose the correct answer from the options given…

Concept: For two positive integers a and b, the HCF (highest common factor) is the product of the lowest powers of the prime factors common to both numbers, and the LCM (lowest common multiple) is the product of the highest powers of every prime factor appearing in either number. These always satisfy HCF × LCM = a × b. Product and Sum are simply the direct arithmetic operations a × b and a + b.

Working:

1. Prime-factorise both numbers: 84 = 22 × 3 × 7 and 96 = 25 × 3.

2. HCF = product of the lowest powers of the common primes (2 and 3) = 22 × 3 = 12.

3. LCM = product of the highest powers of every prime involved (2, 3, 7) = 25 × 3 × 7 = 672.

4. Product = 84 × 96 = 8064.

5. Sum = 84 + 96 = 180.

Cross-check: For any two numbers, HCF × LCM must equal their Product. Here 12 × 672 = 8064, which matches the directly computed Product 84 × 96 = 8064 — confirming both the HCF and LCM values.

Result: List I matches List II as A - IV, B - I, C - II, D - III.