Math Tools
Prime Factorization Calculator
Break a whole number into its prime factors in exponent form and expanded multiplication form.
Prime Factorization Calculator
Break a number into its prime factors and view the result in exponent form and expanded multiplication form.
Results
Prime factorization
2^3 × 3^2 × 5
Expanded prime factors
2 × 2 × 2 × 3 × 3 × 5
Prime number?
No
Prime powers
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What this calculator does
This calculator breaks a whole number into its prime factors and shows both exponent form and expanded multiplication form. It can help you check homework, simplify number problems, and understand how a number is built from prime numbers.
What is prime factorization?
Prime factorization means expressing a whole number as a product of prime numbers. A prime number is a number greater than 1 that can be divided only by 1 and itself, such as 2, 3, 5, 7, and 11.
For example, the prime factorization of 360 is 2^3 × 3^2 × 5. This tells you exactly which prime numbers multiply together to make 360. Prime factorization is widely used in arithmetic, algebra, number theory, greatest common factor problems, least common multiple problems, and fraction simplification.
How to do prime factorization step by step
Start with the number you want to factor. Divide it by the smallest prime number possible. If the result can still be divided by a prime number, keep going until the quotient becomes 1.
For example, to factor 360, divide by 2 to get 180, divide by 2 again to get 90, divide by 2 again to get 45, then divide by 3 to get 15, divide by 3 to get 5, and divide by 5 to get 1. The prime factors are 2 × 2 × 2 × 3 × 3 × 5, or 2^3 × 3^2 × 5.
Prime factorization examples
| Number | Prime factorization | Expanded form |
|---|---|---|
| 24 | 2^3 × 3 | 2 × 2 × 2 × 3 |
| 36 | 2^2 × 3^2 | 2 × 2 × 3 × 3 |
| 60 | 2^2 × 3 × 5 | 2 × 2 × 3 × 5 |
| 84 | 2^2 × 3 × 7 | 2 × 2 × 3 × 7 |
| 100 | 2^2 × 5^2 | 2 × 2 × 5 × 5 |
| 360 | 2^3 × 3^2 × 5 | 2 × 2 × 2 × 3 × 3 × 5 |
Prime factorization table for common numbers
This quick prime factorization table gives common examples people often search for when checking homework, simplifying fractions, or reviewing number properties.
| Number | Prime factorization |
|---|---|
| 12 | 2^2 × 3 |
| 18 | 2 × 3^2 |
| 20 | 2^2 × 5 |
| 48 | 2^4 × 3 |
| 72 | 2^3 × 3^2 |
| 90 | 2 × 3^2 × 5 |
| 120 | 2^3 × 3 × 5 |
| 144 | 2^4 × 3^2 |
| 180 | 2^2 × 3^2 × 5 |
| 210 | 2 × 3 × 5 × 7 |
Prime number reference table
Prime factorization depends on knowing which numbers are prime and which numbers are composite. This quick reference helps explain the difference.
| Number type | Meaning |
|---|---|
| 1 | Not prime and not composite |
| 2 | Smallest prime number and only even prime |
| 3, 5, 7 | Common small prime factors |
| 11, 13, 17, 19 | Prime numbers often used in factorization practice |
| Composite numbers | Can be broken into smaller prime factors |
Why prime factorization is important
Prime factorization helps you understand the structure of numbers. It is one of the main tools used to find the greatest common factor and least common multiple of two or more numbers.
It is also helpful when simplifying fractions, solving divisibility problems, comparing number patterns, and checking whether a number is prime or composite. Students often use prime factorization in middle school and early algebra, while more advanced math uses it in deeper number theory topics.
Prime numbers and composite numbers explained
A prime number has exactly two positive factors: 1 and itself. That means numbers like 2, 3, 5, 7, and 13 are prime. A composite number has more than two positive factors, such as 12, 18, 24, and 100.
Prime factorization only breaks composite numbers into smaller prime pieces. If the input number is already prime, then its prime factorization is the number itself.
Where a prime factorization calculator is useful
A prime factorization calculator is useful for students, teachers, tutors, and anyone solving arithmetic or algebra problems. It gives a fast way to check answers and understand how a number can be built from prime factors.
It is especially useful when working on greatest common factor, least common multiple, reducing fractions, factoring integers, and identifying whether a number is prime or composite.
Prime factorization calculator FAQ and common questions
What is prime factorization?
Prime factorization is writing a whole number as a product of prime numbers. For example, 60 = 2^2 × 3 × 5.
How do you find the prime factorization of a number?
Divide the number by the smallest prime number possible, continue factoring each quotient, and stop when the result becomes 1.
What is the prime factorization of 360?
The prime factorization of 360 is 2^3 × 3^2 × 5, or in expanded form, 2 × 2 × 2 × 3 × 3 × 5.
Can a prime number be written in prime factorization form?
Yes. A prime number factorizes to itself. For example, the prime factorization of 17 is simply 17.
Why is prime factorization useful for GCF and LCM?
Prime factorizations make it easier to compare the prime building blocks of numbers, which helps when finding common factors and common multiples.
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