Prime factorization sits at the heart of number theory. The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has a unique prime factorization โ the irreducible building blocks from which all other numbers are assembled.
Trial Division and the Sieve of Eratosthenes
Prime factorization sits at the heart of number theory. The Fundamental Theorem of Arithmetic guarantees that every integer greater than 1 has a unique prime factorization โ the building blocks from which all other numbers are assembled. The most direct method is trial division: divide the number by 2, then 3, 5, 7, and so on up to the square root of n. Each successful division yields a prime factor, and the exponents count how many times each prime appears.
GCF, LCM, and Simplifying Fractions with Prime Factors
The GCD and LCM are intimately related through prime factorizations. The GCD takes the minimum exponent of each shared prime; the LCM takes the maximum. The Euclidean algorithm computes the GCD far more efficiently than full factorization โ it runs in O(log min(a, b)) steps regardless of how large the numbers are. The Sieve of Eratosthenes is the classic algorithm for finding all primes up to a limit N. Starting from 2, it marks all multiples of each prime as composite. The remaining unmarked numbers are prime. The sieve runs in O(N log log N) time and is practical up to tens of millions.
Cryptography, RSA Encryption, and Why Large Primes Matter
Prime factorization underpins RSA encryption: factoring the product of two large primes is computationally infeasible with current algorithms, providing the security of public-key cryptography.