Prime Factorization Calculator
Factor Tree & Exponent Notation
Understanding Prime Factorization
Prime numbers are the building blocks of every whole number — and prime factorization is the process of finding exactly which primes multiply together to build any given number.
Every whole number greater than 1 has exactly one prime factorization, no matter how you break it down — this uniqueness is one of the cornerstones of number theory.
Instead of writing out a repeated prime many times, exponent notation collapses it into a base and a power — 2 × 2 × 2 becomes 2³, keeping the notation short and easy to scan.
Worked Example
Breaking down 360: 360 ÷ 2 = 180, 180 ÷ 2 = 90, 90 ÷ 2 = 45, 45 ÷ 3 = 15, 15 ÷ 3 = 5, and 5 is already prime. Collecting the primes used gives 2 × 2 × 2 × 3 × 3 × 5, or in exponent notation, 360 = 2³ × 3² × 5.
Key Takeaways
- Every whole number greater than 1 has a unique prime factorization, regardless of the order the primes are found in.
- Prime factorization is the foundation for finding the LCM and GCF of two or more numbers.
- A prime number's own prime factorization is just itself, since it has no smaller prime factors to divide out.
Frequently Asked Questions
- Enter any whole number greater than 1.
- Click "Calculate."
- See the number broken down step by step into its prime factors, along with the compact exponent notation, such as
360 = 2³ × 3² × 5.
Prime factorization is the process of breaking a number down into the set of prime numbers that multiply together to produce it. Every whole number greater than 1 has exactly one prime factorization (ignoring the order of the factors) — this is known as the Fundamental Theorem of Arithmetic.
When the same prime factor appears more than once, exponent notation groups the repeats together instead of listing them individually. For example, 2 × 2 × 2 × 3 × 3 × 5 is written more compactly as 2³ × 3² × 5, where the small raised number (the exponent) shows how many times that prime is repeated.
Listing all factors of a number (like 1, 2, 3, 4, 6, 8, 12, 24 for the number 24) includes every divisor, including composite ones. Prime factorization only uses prime numbers as building blocks — for 24, that's 2³ × 3 — and every one of those regular factors can be reconstructed by multiplying different combinations of the prime factors together.
If the number you enter is already prime, the calculator tells you directly, since a prime number's only prime factor is itself (for example, the prime factorization of 17 is simply 17).
Prime factorization is only defined for whole numbers greater than 1 — the number 1 has no prime factors at all, and negative numbers and decimals fall outside the definition entirely. The calculator shows a clear error for any of these cases.