Prime factorization
Also known as: prime decomposition
Prime factorization is the process of writing a whole number as a product of prime numbers. Every integer greater than 1 has exactly one prime factorization — for example, 60 = 2² × 3 × 5.
Prime factorization breaks a whole number down into the prime numbers that multiply to produce it. A prime is a number greater than 1 whose only factors are 1 and itself (2, 3, 5, 7, 11, ...). The prime factorization of 60 is 2 × 2 × 3 × 5, usually written with exponents as 2² × 3 × 5.
The standard method is a factor tree: split the number into any two factors, then keep splitting each branch until every leaf is prime. Starting 60 as 6 × 10, then 6 = 2 × 3 and 10 = 2 × 5, leaves the primes 2, 2, 3, 5. A key fact — the fundamental theorem of arithmetic — guarantees that no matter how you split along the way, every whole number greater than 1 has exactly one prime factorization.
Prime factorization is the workhorse behind many other skills. It gives the greatest common factor (take the shared primes at the lowest powers) and the least common multiple (take all primes at the highest powers), simplifies fractions and radicals — √60 = √(2² × 15) = 2√15 — and answers divisibility questions, such as counting how many factors a number has from the exponents in its factorization.
Standardized tests lean on prime factorization constantly. The ACT tests it directly in intermediate algebra, the GRE builds divisor, multiple, and remainder problems on it in quantitative reasoning, and the Praxis Core math exam includes it among properties of whole numbers. If you can factor quickly and accurately, a whole family of number-theory questions becomes routine.
Key takeaways
- Prime factorization expresses a number as a product of primes, e.g. 60 = 2² × 3 × 5.
- Every whole number greater than 1 has exactly one prime factorization (the fundamental theorem of arithmetic).
- Factor trees are the standard technique: keep splitting until every branch ends in a prime.
- Prime factorizations give GCF, LCM, simplified radicals, and factor counts.
- The ACT, GRE, and Praxis Core math exams all test prime factorization and its applications.