Prime Factorization Fundamentals: The Fundamental Theorem of Arithmetic
Prime factorization is the mathematical decomposition of a composite positive integer into a unique product of prime numbers (integers greater than 1 with no positive divisors other than 1 and themselves).
The Fundamental Theorem of Arithmetic states that every integer n > 1 either is a prime number itself or can be represented uniquely as a product of prime powers, up to the order of factors: n = p_1^(e_1) · p_2^(e_2) ⋯ p_k^(e_k) = ∏_(i=1)^(k) p_i^(e_i) where p_1 < p_2 < … < p_k are distinct prime numbers and e_i ≥ 1 are integer exponents.
For example, the prime factorization of 360 is: 360 = 2^3 · 3^2 · 5^1 = 8 · 9 · 5 No other set of prime numbers can multiply to produce 360. Prime factorization provides the foundational building blocks for number theory, modular arithmetic, algebra, fraction reduction, and modern public-key cryptography (RSA encryption).