Prime Factorization Calculator
Break any whole number into its prime factors, see the division steps, and get the exponent form and all its factors.
Results
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Steps
| Step | Working |
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Quick Answer
Prime factorization writes a number as a product of prime numbers. Divide by the smallest prime that fits, and keep going until only 1 is left. 360 ÷ 2 ÷ 2 ÷ 2 ÷ 3 ÷ 3 ÷ 5 = 1, so 360 = 2 × 2 × 2 × 3 × 3 × 5, or 2^3 × 3^2 × 5.
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Table of Contents
- Prime Factorization Calculator
- Quick Answer
- Related Calculators
- How This Prime Factorization Calculator Works
- How to Use the Prime Factorization Calculator
- Core Rules
- Key Terms
- Worked Example
- Real-Life Examples
- Prime Numbers vs Composite Numbers
- Prime Factorizations of Common Numbers
- Why Prime Factors Are Useful
- Common Mistakes to Avoid
- Frequently Asked Questions
- References & Sources
How This Prime Factorization Calculator Works
Enter a whole number. The calculator divides it by 2 as many times as it can, then by 3, 5, 7 and so on, until what is left is 1 or a prime. Each division is shown as a step. It also counts how many factors the number has and lists them.
Quick explanation: every whole number above 1 is either prime or a unique product of primes.
- Exponent form: repeated primes are written as powers, such as 2³.
- Number of factors: add 1 to each exponent and multiply.
- All factors: listed when there are 64 or fewer.
How to Use the Prime Factorization Calculator
- Enter a whole number from 2 to one trillion.
- Press Calculate.
- Read the prime factorization in exponent form.
- See how many factors the number has, and the full list for smaller counts.
- Follow the division steps to see how the factors were found.
Core Rules
- Prime number: a whole number above 1 with exactly two factors, 1 and itself
- Method: divide by the smallest prime that fits, and repeat
- Stopping point: you only need to test divisors up to the square root
- Number of factors: for p^a × q^b, it is (a + 1) × (b + 1)
Key Terms
- Prime number: a whole number above 1 that only divides by 1 and itself.
- Prime factor: a prime that divides a number exactly.
- Factor tree: a diagram that splits a number into factors until all are prime.
- Composite number: a whole number above 1 that is not prime.
- Exponent form: writing repeated factors as powers, such as 2³ for 2 × 2 × 2.
Worked Example
Find the prime factors of 84.
- 84 ÷ 2 = 42
- 42 ÷ 2 = 21
- 21 ÷ 3 = 7, and 7 is prime
- 84 = 2² × 3 × 7, with (2 + 1) × (1 + 1) × (1 + 1) = 12 factors
Real-Life Examples
- Simplifying a fraction: 84/126 = (2² × 3 × 7) / (2 × 3² × 7). Cancel 2 × 3 × 7 to get 2/3.
- Finding the GCF: 48 = 2^4 × 3 and 180 = 2^2 × 3^2 × 5. The shared lowest powers are 2² × 3 = 12.
- Simplifying a square root: 72 = 2³ × 3², so √72 = 2 × 3 × √2 = 6√2.
- Checking for a prime: 97 has no factors except 1 and 97, so it is prime.
Prime Numbers vs Composite Numbers
- Prime: exactly two factors, such as 2, 3, 5, 7 and 11.
- Composite: more than two factors, such as 4, 6, 8, 9 and 12.
- 1: neither prime nor composite.
- 2: the only even prime number.
Prime Factorizations of Common Numbers
| Number | Prime factorization | Number of factors |
|---|---|---|
| 12 | 2^2 × 3 | 6 |
| 24 | 2^3 × 3 | 8 |
| 36 | 2^2 × 3^2 | 9 |
| 48 | 2^4 × 3 | 10 |
| 60 | 2^2 × 3 × 5 | 12 |
| 72 | 2^3 × 3^2 | 12 |
| 100 | 2^2 × 5^2 | 9 |
| 120 | 2^3 × 3 × 5 | 16 |
| 144 | 2^4 × 3^2 | 15 |
| 360 | 2^3 × 3^2 × 5 | 24 |
| 1,000 | 2^3 × 5^3 | 16 |
Why Prime Factors Are Useful
- LCM and GCF: both come straight from prime factors.
- Simplifying fractions: cancel shared prime factors.
- Square roots: pairs of primes come out of the root, such as √72 = 6√2.
- Encryption: online security relies on how hard it is to factor very large numbers.
Common Mistakes to Avoid
- Calling 1 a prime. 1 has only one factor, so it is not prime.
- Stopping at a composite number. Keep dividing until every factor is prime.
- Skipping small primes. Always try 2, then 3, then 5, in order.
- Forgetting repeated factors. 8 = 2 × 2 × 2, not just 2.
- Testing too far. You only need to test divisors up to the square root.
Frequently Asked Questions
2^3 × 3^2 × 5.
2² × 5².
No. A prime has exactly two factors, and 1 has only one.
Divide by the smallest prime that fits, write it down, and repeat with the result until you reach 1.
Yes. This is the fundamental theorem of arithmetic.
A diagram where you split a number into two factors, then split those, until every branch ends in a prime.
24, because 360 = 2³ × 3² × 5 and (3 + 1) × (2 + 1) × (1 + 1) = 24.
No. 91 = 7 × 13.
The record keeps changing. The largest known primes are Mersenne primes of the form 2^p − 1, with millions of digits.
References & Sources
| Source | Used For | Link |
|---|---|---|
| Wikipedia, Integer factorization | Methods for finding prime factors. | https://en.wikipedia.org/wiki/Integer_factorization |
| Wikipedia, Fundamental theorem of arithmetic | Every number has a unique prime factorization. | https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic |