Prime Factorisation Calculator
Enter a whole number greater than 1 to break it into its prime factors. The calculator shows the expanded factorisation, index form, factorisation steps and number of positive factors.
Enter a whole number
360 is a prime number — its only factors are 1 and itself.
Expanded form: 2 × 2 × 2 × 3 × 3 × 5
| Step | Value |
|---|---|
| Start with | 360 |
| Divide by 2 | 360 ÷ 2 = 180 |
| Divide by 2 | 180 ÷ 2 = 90 |
| Divide by 2 | 90 ÷ 2 = 45 |
| Divide by 3 | 45 ÷ 3 = 15 |
| Divide by 3 | 15 ÷ 3 = 5 |
| Remaining number | 5 is prime |
| Prime factors | 2 × 2 × 2 × 3 × 3 × 5 |
| Index form | 2³ × 3² × 5 |
| Step | Value |
|---|---|
| Prime factorisation | 2³ × 3² × 5¹ |
| Add 1 to each exponent | (3 + 1) × (2 + 1) × (1 + 1) |
| Multiply | 4 × 3 × 2 |
| Positive factors | 24 |
How to use this calculator
- Enter a whole number greater than 1. Grouping commas such as
1,000,000are fine. - Read the result: the prime factorisation in index form is the large figure, with the expanded form, whether the number is prime or composite, its distinct prime factors and its number of positive factors beside it.
- The first table shows every division used to factorise the number; the second shows how the number of positive factors is worked out.
- Use Reset to return to the starting number.
The calculation runs in the page as you type — nothing is sent to a server.
What is prime factorisation?
Prime factorisation writes a whole number as a multiplication of prime numbers. Repeated factors are usually collected into index form using powers.
- 2, 3 and 5 are the prime factors of 60.
- 2 appears twice, so it is written as 2².
- Every whole number greater than 1 has one unique prime factorisation, apart from the order of its factors.
What is a prime number?
A prime number has exactly two positive factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13 and 17. If the number you enter is prime, its prime factorisation is the number itself — for example 97 = 97. A prime is not a number that "cannot be factorised"; its factorisation is simply a single prime.
How is the number of positive factors worked out?
Once a number is in index form, add 1 to each exponent and multiply the results. Every divisor of the number is made by choosing how many of each prime to include, from none up to its exponent.
- Each exponent e is how many times that prime appears.
- The count includes 1 and the number itself.
Worked example: 360
360 is even, so divide by 2 repeatedly, then move to the next prime. The number left at the end, 5, is prime.
| Step | Value |
|---|---|
| Start with | 360 |
| Divide by 2 | 360 ÷ 2 = 180 |
| Divide by 2 | 180 ÷ 2 = 90 |
| Divide by 2 | 90 ÷ 2 = 45 |
| Divide by 3 | 45 ÷ 3 = 15 |
| Divide by 3 | 15 ÷ 3 = 5 |
| Remaining number | 5 is prime |
| Prime factors | 2 × 2 × 2 × 3 × 3 × 5 |
| Index form | 2³ × 3² × 5 |
So 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5. The number of positive factors is:
| Step | Value |
|---|---|
| Prime factorisation | 2³ × 3² × 5¹ |
| Add 1 to each exponent | (3 + 1) × (2 + 1) × (1 + 1) |
| Multiply | 4 × 3 × 2 |
| Positive factors | 24 |
360 has three distinct prime factors (2, 3 and 5), six prime factors counting repeats, and 24 positive factors in total.
What this calculator covers
It factorises one positive whole number from 2 up to 1,000,000,000 using trial division, and shows the working. It does not draw a factor tree, take decimals or negative numbers, or list every divisor — only the count of them. Decimals are rejected rather than rounded, so12.9 is not treated as 13.