Fundamental Theorem Of Arithemetic states that every composite number is a product of prime number ..example --12 it can be expressed as 2*2*3...

**Fundamental Theorem of Arithmetic**

The Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be represented uniquely as a product of prime numbers, up to the order of the factors.

**Explanation:**

1. **Prime Numbers:**
- Prime numbers are positive integers greater than 1 that have no divisors other than 1 and themselves. Examples of prime numbers include 2, 3, 5, 7, 11, and so on.

2. **Prime Factorization:**
- Prime factorization is the process of expressing a composite number as a product of prime numbers.
- According to the Fundamental Theorem of Arithmetic, every composite number can be expressed as a unique product of prime factors.
- For example, let's consider the number 24. Its prime factorization is 2 × 2 × 2 × 3 or 2^3 × 3.

3. **Uniqueness of Prime Factorization:**
- The Fundamental Theorem of Arithmetic states that the prime factorization of a number is unique, meaning there is only one way to express a number as a product of prime factors.
- For example, if we consider the number 24, its prime factorization is 2 × 2 × 2 × 3. No matter how we rearrange the factors or replace them with different prime numbers, we will always end up with the same result.

4. **Applications:**
- The Fundamental Theorem of Arithmetic is widely used in various mathematical concepts and applications.
- It is crucial in number theory, cryptography, and prime number generation.
- It helps in simplifying fractions and finding the greatest common divisor (GCD) and least common multiple (LCM) of two or more numbers.
- The theorem also plays a significant role in determining whether a number is prime or composite.

In conclusion, the Fundamental Theorem of Arithmetic states that every positive integer greater than 1 can be expressed uniquely as a product of prime numbers, up to the order of the factors. This theorem has various applications in mathematics and is essential for understanding number theory and related concepts.