Prime factorization is the process of breaking down a number into the prime numbers that multiply together to give the original number. For example, the prime factorization of 12 is 2 × 2 × 3. On the other hand, finding prime numbers involves checking whether a number is only divisible by 1 and itself. The Sieve of Eratosthenes is a common algorithm for generating all primes up to a given limit, while trial division is used for checking if a single number is prime. Both concepts are important in competitive programming, especially in problems involving divisors, modular arithmetic, or number theory.