Let smaller odd number = x.

Then x+(x+2)=56.

So 2x+2=56, 2x=54, x=27.

Check: 27+29=56 ✓

A number has exactly 3 factors only when it is the square of a prime.

For p² where p is prime, factors are: 1, p, p².

Example: 4 has factors 1,2,4 (3 factors). 9 has factors 1,3,9 (3 factors).

Divisors of 28: 1, 2, 4, 7, 14, 28.

Sum = 1+2+4+7+14+28 = 56. (Note: 28 is a perfect number where sum of proper divisors = 28)

For n = p₁^a × p₂^b × p₃^c, number of divisors = (a+1)(b+1)(c+1).

Here: (3+1)(2+1)(1+1) = 4×3×2 = 24

Using prime factorization: 48 = 2⁴ × 3, 64 = 2⁶. HCF = 2⁴ = 16 (taking lowest powers of common prime factors).

Prime factorization: 12 = 2² × 3, 18 = 2 × 3². LCM = 2² × 3² = 4 × 9 = 36 (taking highest powers of all prime factors).

This question asks us to find the Highest Common Factor (HCF) of two numbers using prime factorization or the Euclidean algorithm.

Express 56 as a product of prime numbers.

Express 72 as a product of prime numbers.

The HCF is the product of common prime factors with their lowest powers.

The HCF of 56 and 72 is 8, making the correct answer (B).

This question asks us to find the Least Common Multiple (LCM) of two numbers using prime factorization.

Break 15 into its prime factors.

Break 25 into its prime factors.

The LCM is found by taking the highest power of each prime that appears in either factorization: 3¹ and 5².

The LCM of 15 and 25 is 75.

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.