Prime Factorization Calculator
Break any whole number into its prime factors, with the divisor count, divisor sum, and a primality check.
Details
A whole number from 2 to 1,000,000,000,000.
Prime factorization of 360
2³ × 3² × 5
Built from 3 distinct primes
How many divisors
24
All divisors added up
1,170
Is it prime?
No
Biggest prime factor
5
This breaks a number down into the prime numbers that multiply together to make it, shown in exponent form.
It also gives every divisor, how many there are, and whether the number is prime.
What prime factorisation is
Prime factorisation breaks a number into the prime numbers that multiply to make it. 360 breaks down to 2³ × 3² × 5.
A prime is a whole number above 1 with no divisors except 1 and itself: 2, 3, 5, 7, 11, 13 and so on. They are the building blocks, because they cannot be broken down further.
Every whole number above 1 has exactly one prime factorisation. Not roughly one, exactly one, in one order. That guarantee is called the fundamental theorem of arithmetic, and it is why the technique is so useful.
Divide by the smallest prime that fits, over and over, until you reach 1. The order you find them in does not matter; the final set is always identical.
What to enter
- Number (n)
- Any whole number above 1. Very large numbers take longer, which is the whole basis of modern encryption.
Quick divisibility checks
- By 2
- The last digit is even.
- By 3
- The digits add up to a multiple of 3. For 360: 3+6+0 = 9, so yes.
- By 5
- It ends in 0 or 5.
- By 9
- The digits add to a multiple of 9.
- By 11
- Alternately add and subtract the digits; if the result is 0 or a multiple of 11, it divides.
What this assumes
1 is not prime, by definition. Allowing it would break the guarantee of a unique factorisation, since you could add as many 1s as you liked.
How to calculate the prime factors of a number
Keep dividing by the smallest prime that goes in, until nothing is left but 1.
- p
- Each distinct prime factor
- a, b, c
- How many times each one appears
Start with the smallest prime. Try 2 first. Keep dividing by 2 while the result stays whole.
Move up when it stops working. Then try 3, then 5, then 7, and so on through the primes.
Stop at 1. When the division reaches 1, you have every factor.
Write it with exponents. Group repeats: 2 × 2 × 2 becomes 2³.
See a worked example: factorising 360
- Number
- 360
360 ÷ 2 = 180. 180 ÷ 2 = 90. 90 ÷ 2 = 45. That is three 2s, and 45 is odd so 2 stops working.
45 ÷ 3 = 15. 15 ÷ 3 = 5. That is two 3s.
5 ÷ 5 = 1. One 5, and we have reached 1.
So 360 = 2³ × 3² × 5. Counting divisors from that: (3+1)(2+1)(1+1) = 24 of them.
360 = 2³ × 3² × 5
Frequently asked questions
- Why is 1 not a prime number?
Because it would break the rule that every number has exactly one prime factorisation. If 1 were prime, 6 could be 2 × 3, or 1 × 2 × 3, or 1 × 1 × 2 × 3, endlessly.
Excluding 1 keeps the factorisation unique, which is what makes it useful. The definition was chosen to preserve that, not the other way round.
- How do I count the divisors from a factorisation?
Add one to each exponent and multiply those together. For 360 = 2³ × 3² × 5, that is (3+1) × (2+1) × (1+1) = 24 divisors.
It works because each divisor is built by choosing how many of each prime to use: 0 to 3 twos, 0 to 2 threes, 0 or 1 five.
- What is prime factorisation used for?
Immediately, for finding the greatest common factor and lowest common multiple, which is how fractions get simplified and added.
More widely, it underpins encryption. Multiplying two large primes is quick, but factorising the result back is extremely slow, and that asymmetry is what protects most online transactions.
- How can I tell if a number is prime?
Test division by every prime up to its square root. If none divide it evenly, it is prime.
You only need to go to the square root because factors come in pairs: if a number has a factor above its square root, it must have a matching one below.
- Is there a largest prime number?
No. Euclid proved over two thousand years ago that primes go on forever, and the proof is short enough to follow in a minute.
The largest known prime keeps growing as computers find bigger ones, but there is always another beyond it.
Problems people actually run into
Stopping before the factors are fully prime
Writing 360 as 8 × 45 is a correct factorisation but not a prime one, because neither 8 nor 45 is prime.
Keep going until every number left is prime: 8 becomes 2³ and 45 becomes 3² × 5. The check is simple — if any factor can still be divided, you are not finished.
Results are estimates for general information only and are not professional financial, medical, or legal advice. Read our full disclaimer.
Last updated: September 4, 2026
