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 this works: Method, 1 source, Checked against 1 worked example,
How this works
Method
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. Full method
How it’s tested
One worked example for this page is checked by automated tests before every release: given the inputs, the tool must show the expected answer.
Privacy
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Changes
- First published, with ten calculators: GCF, LCM, factors, prime checker, prime factorization, mean, median and mode, standard deviation, variance, exponents and logarithms.
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.