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Product of two numbers= Product of their HCF & LCM

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The Unique Factorization Theorem (UFT), also known as the Fundamental Theorem of :

Define as Every positive integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors.

Ⓜ️athematical Formulation:

For any positive integer n > 1, there exists a unique set of prime numbers p1, p2, ..., pk and non-negative integers a1, a2, ..., ak such that:

n = p1^a1 × p2^a2 × ... × pk^ak

where p1, p2, ..., pk are distinct prime numbers.

Key Points:

1. Uniqueness: The factorization is unique, except for the order of the factors.
2. Existence: Every positive integer has a prime factorization.
3. : Only prime numbers are used in the factorization.

Examples:

1. 12 = 2^2 × 3
2. 25 = 5^2
3. 36 = 2^2 × 3^2
4. 48 = 2^4 × 3

Consequences:

1. Greatest Common Divisor (GCD) calculation
2. Least Common Multiple ( ) calculation
3. Prime number theory
4. Number theory and cryptography
5. Algebra and group theory

Proof:

The proof involves:

1. Existence: Using mathematical induction to show every integer has a prime factorization.
2. Uniqueness: Showing that if two factorizations exist, they must be identical.

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