The Fundamental Theorem of Arithmetic states that:

Every integer greater than 1 can be written uniquely (up to the order of the factors) as a product of prime numbers.

In simpler terms:

• Any integer $n > 1$ is either a prime number itself or can be factored into a product of primes.
• This factorization is unique, meaning that regardless of how you factor a number, the prime factors (and their powers) will always be the same, though the order of multiplication may vary.

Example:

Consider the number 60.

• 60 can be factored as $60 = 2^2 \times 3 \times 5$.
• No matter how you factor 60, you'll always end up with the prime factors 2, 3, and 5.

Why is it important?

The Fundamental Theorem of Arithmetic is crucial because it ensures the uniqueness of prime factorization, which is the foundation for many areas of number theory, cryptography, and algebra.

Would you like to explore any details or have any questions?

Here are 5 related questions to expand on this topic:

1. How is the Fundamental Theorem of Arithmetic applied in cryptography?
2. Can a negative integer be factored using the Fundamental Theorem of Arithmetic?
3. How do you find the prime factorization of a large number?
4. What role do prime numbers play in modern mathematics?
5. How does the Fundamental Theorem of Arithmetic relate to the Euclidean Algorithm?

Tip: When factoring numbers, it's often easier to start with the smallest prime (2) and work your way up through larger primes!