About the Prime Factorization Calculator
Prime factorisation breaks a number into the primes that multiply to produce it. Every integer above 1 has exactly one such decomposition, which makes it the canonical fingerprint of a number and the foundation for finding common factors and multiples.
The formula
n = p₁^a₁ · p₂^a₂ · … · pₖ^aₖEach p is a distinct prime and each a is how many times it appears. For 360 the factorisation is 2³ × 3² × 5.
How to use this calculator
- 1Enter your Number. The field starts at
360, which you can overwrite. - 2Read the result straight away — it recalculates as you type, so there is no button to press. Use Share to copy a link that reopens the page with your exact numbers filled in.
Worked example
| Input | Value |
|---|---|
| Number | 360 |
Result
Prime Factorization: 2^3 × 3^2 × 5
Prime Factors: 2, 3, 5
Number of Divisors: 24
Largest Prime Factor: 5
Those are the values the page loads with, so you can reproduce this result yourself and then change one field at a time to see what drives the outcome.
Understanding your result
Trial division is the straightforward method: divide by 2 as often as possible, then 3, then 5, and so on upward, recording each success. You can stop once your divisor exceeds the square root of what remains, because anything left at that point must itself be prime.
Uniqueness — the fundamental theorem of arithmetic — is what gives factorisation its power. Because there is only one prime decomposition, the greatest common divisor of two numbers is found by taking the lowest power of each shared prime, and the least common multiple by taking the highest power of every prime that appears. Both become mechanical once the factorisations are in hand.
Things worth knowing
- Always begin with the smallest primes. Dividing out all the 2s first simplifies everything that follows.
- You only need to test primes up to the square root of the remaining value.
- Exponent notation is the conventional form: 360 = 2³ × 3² × 5, not a long list of repeated factors.
- Perfect squares have all-even exponents, which is a quick way to recognise them.
- Factoring large numbers is computationally hard, and that difficulty is deliberate in cryptography rather than a limitation.
Frequently asked questions
What is prime factorisation?+
Expressing a number as a product of prime numbers. 360 becomes 2³ × 3² × 5. Every integer greater than 1 has exactly one such factorisation, ignoring the order of the factors.
How do I factor a number by hand?+
Divide repeatedly by the smallest prime that fits, moving up through 2, 3, 5, 7 and so on. Record each divisor and continue with the quotient until you reach 1.
Why is prime factorisation useful?+
It makes greatest common divisors and least common multiples straightforward, simplifies fractions and radicals, and is central to number theory. It is also the hard problem underpinning RSA encryption.
What is the prime factorisation of a prime number?+
The number itself. A prime has no factors other than 1 and itself, so its factorisation is a single term — 17 factorises as 17.