Prime Factorization Calculator
Type a whole number to split it into primes, one division at a time, and see it written as a product of prime powers.
Any whole number up to 20 digits.
Prime factors of 360
2³ × 3² × 5
360 = 2³ × 3² × 5, a product of 6 primes (3 different).
- Expanded
- 2 × 2 × 2 × 3 × 3 × 5
- Distinct primes
- 3
- Number of factors
- 24
How to find the prime factorization
Divide by the smallest prime that goes in exactly, write it down, and repeat with the quotient until you reach 1. For 360: 360 ÷ 2 = 180, ÷ 2 = 90, ÷ 2 = 45; 2 no longer divides, so try 3: 45 ÷ 3 = 15, ÷ 3 = 5; then 5 ÷ 5 = 1. The primes you divided by are the factorization: 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5.
A factor tree does the same thing with any split you like: 360 = 36 × 10 = (6 × 6) × (2 × 5) = (2 × 3 × 2 × 3) × 2 × 5. Whichever path you take, you end at the same primes, because every whole number above 1 has exactly one prime factorization (the fundamental theorem of arithmetic).
The factorization is the key to many other answers: the number of factors (multiply each exponent plus one), whether it is a perfect square (all exponents even), the GCF and LCM of several numbers, and simplifying fractions and square roots (√360 = √(2² × 3² × 10) = 6√10).
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ, unique apart from order
- Every integer greater than 1 can be written as a product of primes in exactly one way, apart from the order of the factors. Source: Wikipedia, Fundamental theorem of arithmetic.
Prime factorizations of common numbers
Each number as a product of prime powers, with how many factors that gives it.
| Number | Prime factorization | Expanded | Factors |
|---|---|---|---|
| 12 | 2² × 3 | 2 × 2 × 3 | 6 |
| 18 | 2 × 3² | 2 × 3 × 3 | 6 |
| 24 | 2³ × 3 | 2 × 2 × 2 × 3 | 8 |
| 30 | 2 × 3 × 5 | 2 × 3 × 5 | 8 |
| 36 | 2² × 3² | 2 × 2 × 3 × 3 | 9 |
| 48 | 2⁴ × 3 | 2 × 2 × 2 × 2 × 3 | 10 |
| 60 | 2² × 3 × 5 | 2 × 2 × 3 × 5 | 12 |
| 72 | 2³ × 3² | 2 × 2 × 2 × 3 × 3 | 12 |
| 84 | 2² × 3 × 7 | 2 × 2 × 3 × 7 | 12 |
| 90 | 2 × 3² × 5 | 2 × 3 × 3 × 5 | 12 |
Show all 21 rowsShow fewer
| 96 | 2⁵ × 3 | 2 × 2 × 2 × 2 × 2 × 3 | 12 |
| 100 | 2² × 5² | 2 × 2 × 5 × 5 | 9 |
| 144 | 2⁴ × 3² | 2 × 2 × 2 × 2 × 3 × 3 | 15 |
| 180 | 2² × 3² × 5 | 2 × 2 × 3 × 3 × 5 | 18 |
| 210 | 2 × 3 × 5 × 7 | 2 × 3 × 5 × 7 | 16 |
| 256 | 2⁸ | 2 × 2 × 2 × 2 × 2 × 2 × 2 × 2 | 9 |
| 360 | 2³ × 3² × 5 | 2 × 2 × 2 × 3 × 3 × 5 | 24 |
| 1,000 | 2³ × 5³ | 2 × 2 × 2 × 5 × 5 × 5 | 16 |
| 1,001 | 7 × 11 × 13 | 7 × 11 × 13 | 8 |
| 2,310 | 2 × 3 × 5 × 7 × 11 | 2 × 3 × 5 × 7 × 11 | 32 |
| 9,999 | 3² × 11 × 101 | 3 × 3 × 11 × 101 | 12 |
Frequently Asked Questions
What is the prime factorization of 100?
100 = 2 × 2 × 5 × 5 = 2² × 5². Both exponents are even, which is why 100 is a perfect square (10²).
What is the prime factorization of a prime number?
Just the number itself: 97 = 97. A prime cannot be split further, so its factor tree has a single leaf.
Why is 1001 interesting?
1001 = 7 × 11 × 13. That is why writing a three-digit number twice (such as 123123) always gives a multiple of 7, 11 and 13: 123123 = 123 × 1001.