36 = 2² × 3², 48 = 2⁴ × 3. HCF = 2² × 3 = 12, LCM = 2⁴ × 3² = 144.

Product = 36 × 48 = HCF × LCM = 12 × 144 = 1728.

Using formula: HCF × LCM = Product of two numbers. 8 × 96 = 24 × x. 768 = 24x. x = 32.

100 = 2² × 5², 150 = 2 × 3 × 5², 200 = 2³ × 5².

Common factors: 2¹ × 5² = 2 × 25 = 50.

This question asks us to find the least common multiple (LCM) of two numbers using prime factorization.

Break 84 into its prime factors by dividing by smallest primes.

Break 140 into its prime factors by dividing by smallest primes.

LCM is found by taking the highest power of each prime factor that appears in either number.

The LCM of 84 and 140 is 420, which is option (A).

Using HCF × LCM = Product of two numbers: 15 × 180 = 45 × other number.

Therefore, other number = 2700 ÷ 45 = 60.

144 = 2⁴ × 3². 196 = 2² × 7².

Common prime factors: 2².

Therefore, HCF = 4.

Let numbers be 4k and 5k. HCF = k = 6.

Therefore, numbers are 24 and 30.

The larger number is 30.

56 = 2³ × 7, 84 = 2² × 3 × 7, 140 = 2² × 5 × 7.

Common prime factors: 2 × 7 = 14.

16 = 2⁴, 24 = 2³ × 3, 36 = 2² × 3². LCM = 2⁴ × 3² = 16 × 9 = 144.

HCF × LCM = Product of numbers. 18 × 324 = 36 × other.

Therefore, other = 5832 ÷ 36 = 162.