Prime Factorisation Calculator
Enter a number and this breaks it into the prime numbers it is built from, written out in full and in index form.
Prime factors
2 × 2 × 2 × 3 × 3 × 5
- Prime factors
- 2 × 2 × 2 × 3 × 3 × 5
- In index form
- 2^3 × 3^2 × 5
- How many factors
- 6
- Distinct primes
- 3
- 360 = 2 × 2 × 2 × 3 × 3 × 5.
- Every whole number above 1 breaks into primes in exactly one way, whatever order you find them in. That is the fundamental theorem of arithmetic, and it is why the answer to this question is never ambiguous.
- To do it by hand, divide by the smallest prime that goes in — 2, then 3, then 5, then 7 — and keep going with the answer until you are left with a prime.
About prime factors
A prime number has exactly two factors: itself and one. 2, 3, 5, 7, 11 and 13 are the first few. Every other whole number above one can be built by multiplying primes together, and prime factorisation is the job of finding which.
One is deliberately not prime. It has only a single factor rather than two, and if it counted as prime you could pad any factorisation with as many ones as you liked and the answer would stop being unique.
Uniqueness is the point of the whole topic. Whatever order you find them in, every number breaks into exactly one set of primes — 360 is always 2 × 2 × 2 × 3 × 3 × 5, never anything else. That result is called the fundamental theorem of arithmetic, and it is why questions about prime factors have one right answer.
The practical use comes later, in fractions. The highest common factor and lowest common multiple both fall straight out of the prime factorisations of two numbers, which is what makes cancelling and adding fractions systematic rather than a matter of spotting things.
What it works out
- The prime factors of any number
- Index form, with the powers collected
- Whether the number is itself prime
- The method for doing it by hand
The formula
Divide by the smallest prime that goes in, and repeat with the answer
Start with the smallest prime and keep going until nothing is left but a prime.
360 divided by 2 is 180. Again by 2 is 90. Again by 2 is 45, which is odd, so 2 is finished. Next prime is 3: 45 divided by 3 is 15, and again by 3 is 5. Five is prime, so that is the end.
Collecting them up: 360 = 2 × 2 × 2 × 3 × 3 × 5, or 2³ × 3² × 5 in index form.
You never have to test past the square root of what is left. Once the number you are dividing by, squared, is bigger than the number you are dividing into, whatever remains has to be prime — because any factor below the square root would have been found already, and one above it would need a partner below.
A factor tree is the same method drawn out. Split the number into any two factors, keep splitting, and the numbers at the ends of the branches are the primes. Different starting splits give the same set at the bottom, which is the uniqueness result in visible form.
- Prime
- A number with exactly two factors — itself and one. Two is the only even one.
- Index form
- The same factors with repeats collected as powers: 2 × 2 × 2 becomes 2³.
A worked example
The prime factors of 360.
That works out to 2 × 2 × 2 × 3 × 3 × 5 .
- Prime factors
- 2 × 2 × 2 × 3 × 3 × 5
- In index form
- 2^3 × 3^2 × 5
- How many factors
- 6
- Distinct primes
- 3
Questions
What are the prime factors of 360?
2 × 2 × 2 × 3 × 3 × 5, which is 2³ × 3² × 5 in index form.
How do you find prime factors?
Divide by the smallest prime that goes in, then keep dividing the answer the same way. Start at 2, move to 3, then 5, then 7, and stop when what is left is prime.
Is 1 a prime number?
No. A prime has exactly two factors and one has only a single factor. Excluding it is also what keeps every factorisation unique — otherwise you could add as many ones as you liked.
What is a factor tree?
A diagram of the same method. Split the number into two factors, keep splitting each branch until every end is prime, and read the primes off the bottom. Any starting split gives the same answer.
How do I know when to stop testing?
When the divisor squared is larger than what is left. Any factor smaller than the square root would already have been found, so whatever remains at that point is prime.
What is index form?
The factorisation with repeats collected into powers. 2 × 2 × 2 × 3 × 3 × 5 becomes 2³ × 3² × 5, which is shorter and easier to compare with another number.
What are prime factors used for?
Finding the highest common factor and lowest common multiple, which is what makes simplifying and adding fractions systematic. They also underpin modern encryption, which relies on very large numbers being hard to factorise.
Is 2 the only even prime?
Yes. Every other even number has 2 as a factor as well as itself and one, so it has more than two factors and cannot be prime.