Solution:

The Fundamental Theorem of Arithmetic states that every composite number can be expressed (factorized) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur.

Solution: Solve Yourself.

Solution:

Prime Factorization:

• 12 = 2² × 3
• 15 = 3 × 5
• 21 = 3 × 7

HCF = Product of the smallest power of common primes = 3

LCM = Product of the highest power of all primes = 2² × 3 × 5 × 7 = 420

Solution:

Assume √5 is rational. Then, it can be written as:

√5 = a/b, where a, b are coprime integers and b ≠ 0

Squaring both sides:

5 = a²/b² ⇒ a² = 5b²

This means a² is divisible by 5, so a must also be divisible by 5.

Let a = 5k. Then:

(5k)² = 5b² ⇒ 25k² = 5b² ⇒ b² = 5k²

Thus, b² is divisible by 5, so b must also be divisible by 5.

This contradicts our assumption that a and b are coprime.

Solution:

Assume 3 + 2√5 is rational. Then:

3 + 2√5 = p/q, where p, q are integers and q ≠ 0

Rearranging:

2√5 = p/q – 3 ⇒ √5 = (p – 3q)/2q

Since (p – 3q)/2q is rational, √5 must also be rational.

But we know √5 is irrational (from Q10), leading to a contradiction.

Solution:

Factorize the expression:

7 × 11 × 13 + 13 = 13(7 × 11 + 1) = 13 × (77 + 1) = 13 × 78

Since the number can be expressed as a product of primes other than 1 and itself, it has more than two factors.