Prime Factorization Calculator
Find the prime factorization of a whole number, shown with exponents.
| Prime Power | Prime | Exponent | Value (p^e) | Running Product |
|---|---|---|---|---|
| 2^3 | 2 | 3 | 8 | 8 |
| 3^2 | 3 | 2 | 9 | 72 |
| 5 | 5 | 1 | 5 | 360 |
Multiplying each prime power together in order rebuilds the original number — the running product reaches 360 once every prime factor is included.
About the Prime Factorization Calculator
Prime factorization breaks a number down into the unique set of prime numbers that multiply together to build it. Every whole number greater than 1 has exactly one such breakdown — that uniqueness (the fundamental theorem of arithmetic) is what makes prime factorization such a powerful tool rather than just an academic exercise.
It shows up in middle and high school math as a stepping stone to finding greatest common factors and least common multiples, simplifying radicals, and reducing fractions — and the same technique underlies parts of number theory and cryptography, where the difficulty of factoring very large numbers into their primes is what keeps certain encryption schemes secure.
Because the breakdown is unique, two numbers' prime factorizations can be compared directly to find exactly what they share and what they don't — which is precisely how GCF and LCM get computed under the hood.
How it’s calculated
The standard method is repeated division: start with the smallest prime (2), divide it out of the number as many times as it goes in evenly, then move to the next prime (3, 5, 7…) and repeat, continuing until only 1 remains.
Each prime that was divided out, along with how many times it divided in (its exponent), becomes one term in the final factorization — written as n = p₁^e₁ × p₂^e₂ × ... For example, 360 breaks down to 2³ × 3² × 5, meaning 2×2×2×3×3×5 multiplies back out to exactly 360.
You only need to test prime divisors up to the square root of whatever's left at each step — if nothing up to that point divides evenly, whatever remains is itself prime and becomes the final factor.
Frequently asked questions
What is prime factorization?
It's expressing a whole number as a product of prime numbers — numbers greater than 1 with no divisors other than 1 and themselves. For example, the prime factorization of 60 is 2² × 3 × 5.
Is 1 a prime number?
No. By definition, a prime number has exactly two distinct divisors (1 and itself); 1 only has one divisor (itself), so it's excluded and doesn't appear in any prime factorization.
How is prime factorization used to find GCF and LCM?
The GCF of two numbers is the product of the smallest power of each prime they share; the LCM is the product of the largest power of every prime appearing in either number. Comparing the two numbers' prime factorizations side by side makes both straightforward to read off.
What's the fastest way to find prime factorization by hand?
Repeatedly divide by the smallest prime that still divides in evenly — start with 2, then 3, 5, 7, and so on — writing down each prime you use until you're left with 1. It's slow for very large numbers but reliable for anything typically encountered in coursework.
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