Prime Factorization: Factor Trees, Division Ladders and Divisibility Rules
Break any number into prime factors using factor trees or the ladder method, with divisibility shortcuts and examples of using primes to find the GCF and LCM.
Every whole number greater than 1 is either prime or can be written as a product of primes in exactly one way, apart from the order. This is called the fundamental theorem of arithmetic, and it makes prime factorization a kind of fingerprint for numbers. Once you have it, finding common factors becomes straightforward.
Primes worth knowing
A prime has exactly two factors: 1 and itself. The first primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47. The number 1 is not prime, and 2 is the only even prime.
Divisibility rules
| Divisible by | Rule | Example |
|---|---|---|
| 2 | Last digit is even | 548 |
| 3 | Sum of digits divisible by 3 | 471 (4+7+1 = 12) |
| 5 | Ends in 0 or 5 | 235 |
| 7 | Double the last digit, subtract from the rest; result divisible by 7 | 343 (34 − 6 = 28) |
| 9 | Sum of digits divisible by 9 | 729 (7+2+9 = 18) |
| 11 | Alternating sum of digits divisible by 11 | 2,343 (2−3+4−3 = 0) |
Method 1: factor tree
Split the number into any two factors, then keep splitting until every branch ends in a prime.
360 → 36 × 10 → (6 × 6) × (2 × 5) → (2 × 3) × (2 × 3) × 2 × 5
Collecting the primes: 360 = 2³ × 3² × 5. A different first split, such as 360 = 8 × 45, gives the same final answer.
Method 2: ladder (repeated division)
Divide by the smallest prime that works, write the quotient underneath, and repeat.
- 588 ÷ 2 = 294
- 294 ÷ 2 = 147
- 147 ÷ 3 = 49
- 49 ÷ 7 = 7
- 7 ÷ 7 = 1
So 588 = 2² × 3 × 7².
When to stop testing
You only need to test primes up to the square root of the number. To check whether 97 is prime, test 2, 3, 5 and 7 (since 11² = 121 is already bigger than 97). None divide it, so 97 is prime.
Using factorizations for GCF and LCM
With 360 = 2³ × 3² × 5 and 588 = 2² × 3 × 7²:
- GCF: shared primes at their lowest power: 2² × 3 = 12.
- LCM: all primes at their highest power: 2³ × 3² × 5 × 7² = 17,640.
The GCF calculator shows each number's prime factorization alongside the result, using the notation 2^3 for 2³.
Counting factors
Add 1 to each exponent and multiply. 360 = 2³ × 3² × 5¹ has (3+1)(2+1)(1+1) = 24 factors. Numbers with many factors, like 360, are why it was chosen for degrees in a circle: it divides evenly in so many ways.
Other uses
- Simplifying square roots: √72 = √(2³ × 3²) = 6√2.
- Simplifying fractions: see simplifying fractions with the GCF.
- Very large numbers: for big numbers, the Euclidean algorithm finds the GCF without factoring.
Further reading from official sources
- Factors and multiples (Grade 6 math) – Khan Academy
