How to find prime factors with a factor ladder
A factor ladder finds the same prime factors as a factor tree, but in a neat column. You may also hear it called the ladder method, repeated division or the division method. They are all the same thing.
Divide by the smallest prime that goes in exactly. Write the answer underneath, and keep going until you reach 1. The primes you divided by are the prime factors.
How to do it
- Write the number, and draw a line down its right-hand side.
- Find the smallest prime that divides it exactly. Write it on the right of the line.
- Divide, and write the answer under the number.
- Do the same with the answer. You can use the same prime again.
- Stop when you reach 1. The primes on the right are the prime factors.
Worked examples
What are the prime factors of 84?
| 84 | 2 |
| 42 | 2 |
| 21 | 3 |
| 7 | 7 |
| 1 |
84 = 2 × 2 × 3 × 7.
What are the prime factors of 90?
| 90 | 2 |
| 45 | 3 |
| 15 | 3 |
| 5 | 5 |
| 1 |
2 does not divide 45, so move up to 3. 90 = 2 × 3 × 3 × 5.
What are the prime factors of 3960?
| 3960 | 2 |
| 1980 | 2 |
| 990 | 2 |
| 495 | 3 |
| 165 | 3 |
| 55 | 5 |
| 11 | 11 |
| 1 |
Big numbers work the same way. 3960 = 2 × 2 × 2 × 3 × 3 × 5 × 11.
Tree or ladder?
Both give the same answer. A tree lets you start with any split you like. A ladder always takes the smallest prime first, so it is tidy and easy to check. Use the one your school uses.
Other countries draw it differently
In the United States, China and Japan the primes go on the left of the number, and you stop as soon as the answer is prime, without writing the 1. The primes you find are the same.
Why start with the smallest prime
Going 2, 3, 5, 7 in order means you never miss a prime and never have to go back. Once a prime stops dividing, it will not divide anything further down either.
To check your answer, multiply the primes on the right: 2 × 2 × 3 × 7 = 84.
Where mistakes happen
- Dividing by a number that is not prime. Dividing 84 by 4 leaves a 4 on the right, and 4 is not a prime factor.
- Stopping before 1. If the last number is not 1, divide it by itself to finish.
- Leaving out a repeated prime. 2 goes into 84 twice, so 2 is written twice.
Questions people ask
Why does it end with 1?
The last prime is divided by itself. That puts every prime on the right, so you can read the answer straight down.
Can I use a ladder for two numbers at once?
Yes. Write them side by side, and the same ladder gives the highest common factor and the lowest common multiple.
Is this the same as long division?
No. Each step is a short division by one small prime, which you can usually do in your head.