OneTaskBy Tim King · Updated
Prime Factorisation Calculator
Prime Factors Shown as a Division Ladder in Index Form, Step by Step
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How the Prime Factorisation Calculator works
This prime factorisation calculator writes any whole number as a product of prime factors, such as 360 = 2³ × 3² × 5. It shows the division ladder line by line, the way it is taught in school, and counts how many factors the number has. It handles numbers up to a thousand trillion.
Every whole number above 1 is either prime or can be built by multiplying primes together, and there is only one way to do it apart from the order. That single answer is the prime factorisation, and it is the key to finding highest common factors, lowest common multiples and square roots by hand.
How the prime factorisation calculator works
It uses the division ladder: divide by the smallest prime that goes in exactly, write down the result, and repeat until you reach 1. The primes down the side are the factors. Grouping repeats gives index form, and the formula (a + 1)(b + 1)… on those powers counts every factor.
First example. 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45, ÷ 3 = 15, ÷ 3 = 5, ÷ 5 = 1. So 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5, with (3 + 1)(2 + 1)(1 + 1) = 24 factors.
Second example. 3,600 = 2⁴ × 3² × 5². Every power is even, so it is a perfect square, and halving the powers gives its square root: 2² × 3 × 5 = 60.
How many factors does 600,851,475,143 have? It splits into four different primes, 71 × 839 × 1,471 × 6,857, so it has 2 × 2 × 2 × 2 = 16 factors.
Division ladder for 360: divide by the smallest prime that goes in, until 1 is left
| Step | Number | Divide by prime | Result |
|---|---|---|---|
| 1 | 360 | 2 | 180 |
| 2 | 180 | 2 | 90 |
| 3 | 90 | 2 | 45 |
| 4 | 45 | 3 | 15 |
| 5 | 15 | 3 | 5 |
| 6 | 5 | 5 | 1 |
Prime factorisations of numbers that come up often, with how many factors each has
| Number | Prime factorisation | Number of factors |
|---|---|---|
| 12 | 2² × 3 | 6 |
| 18 | 2 × 3² | 6 |
| 24 | 2³ × 3 | 8 |
| 30 | 2 × 3 × 5 | 8 |
| 36 | 2² × 3² | 9 |
| 40 | 2³ × 5 | 8 |
| 48 | 2⁴ × 3 | 10 |
| 60 | 2² × 3 × 5 | 12 |
| 72 | 2³ × 3² | 12 |
| 84 | 2² × 3 × 7 | 12 |
| 90 | 2 × 3² × 5 | 12 |
| 96 | 2⁵ × 3 | 12 |
| 100 | 2² × 5² | 9 |
| 120 | 2³ × 3 × 5 | 16 |
| 144 | 2⁴ × 3² | 15 |
| 180 | 2² × 3² × 5 | 18 |
| 210 | 2 × 3 × 5 × 7 | 16 |
| 240 | 2⁴ × 3 × 5 | 20 |
| 360 | 2³ × 3² × 5 | 24 |
| 420 | 2² × 3 × 5 × 7 | 24 |
| 720 | 2⁴ × 3² × 5 | 30 |
| 1,000 | 2³ × 5³ | 16 |
| 1,024 | 2¹⁰ | 11 |
| 5,040 | 2⁴ × 3² × 5 × 7 | 60 |
Tips
- Test divisibility quickly. Even numbers take 2; if the digits add to a multiple of 3, so does the number.
- Stop at the square root. If no prime up to √n divides it, what is left is prime.
- A factor tree gives the same answer. Any split works; the primes at the ends always match the ladder.
FAQ
What is the difference between a factor tree and a division ladder?
Both find the same primes. A factor tree splits the number into any two factors and keeps branching until every end is prime, so it can look different each time. A division ladder always divides by the smallest prime first, which keeps the working tidy and the primes in order.
How do you write a prime factorisation in index form?
Group equal primes and write how many there are as a power. 2 × 2 × 2 × 3 × 3 × 5 becomes 2³ × 3² × 5. A prime that appears once has no power written. Exams usually ask for index form, so the calculator shows it first.
How does prime factorisation help find the HCF and LCM?
Write both numbers in index form. The highest common factor takes each shared prime at its lowest power; the lowest common multiple takes every prime at its highest power. For 360 = 2³ × 3² × 5 and 84 = 2² × 3 × 7, the HCF is 2² × 3 = 12 and the LCM is 2³ × 3² × 5 × 7 = 2,520.
Why does 1 have no prime factorisation?
1 is not prime, and multiplying no primes together gives 1 by convention, so there is nothing to list. Keeping 1 out of the primes is what makes every factorisation unique; otherwise 6 could be 2 × 3, 1 × 2 × 3 or 1 × 1 × 2 × 3, and the answer would never be fixed.
See also: all Number Theory tools, Arithmetic, Algebra and Geometry, Statistics, Trigonometry and Calculus.
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