Prime factorization means expressing a positive integer greater than 1 as a product of prime numbers. Prime numbers have exactly two positive divisors: 1 and themselves. Factorization is useful for simplifying fractions, finding the greatest common factor (GCF), and calculating the least common multiple (LCM).
How to use a prime factorization calculator
- Open the Prime Factorization Calculator.
- Enter a positive whole number greater than 1.
- Calculate its prime factors.
- Multiply the reported prime factors together to confirm that they reproduce the original number.
Prime factorization by repeated division
Start with the smallest prime number, 2. Divide the number repeatedly by 2 while it divides evenly. Then test 3, 5, 7, and larger primes as necessary. Continue until the remaining quotient is 1.
Worked example: prime factors of 84
84 ÷ 2 = 42; 42 ÷ 2 = 21; 21 ÷ 3 = 7; and 7 ÷ 7 = 1. Therefore, 84 = 2 × 2 × 3 × 7 = 2² × 3 × 7. Every factor in this final expression is prime.
Worked example: prime factors of 180
180 = 2 × 90 = 2 × 2 × 45 = 2² × 3 × 15 = 2² × 3² × 5. Check: 4 × 9 × 5 = 180.
Common prime factorizations
| Number | Prime factorization |
|---|---|
| 12 | 2² × 3 |
| 24 | 2³ × 3 |
| 36 | 2² × 3² |
| 60 | 2² × 3 × 5 |
| 100 | 2² × 5² |
Using prime factors to find GCF and LCM
Consider 24 = 2³ × 3 and 36 = 2² × 3². The GCF uses the lowest powers shared by both: 2² × 3 = 12. The LCM uses the highest powers of all primes present: 2³ × 3² = 72.
Is 1 a prime number?
No. The number 1 has only one positive divisor, so it is neither prime nor composite. The standard unique prime factorization theorem applies to integers greater than 1.
Common mistakes
- Stopping at a composite factor, such as writing 60 = 6 × 10 as the final prime factorization.
- Calling 1 a prime number.
- Forgetting repeated prime factors.
- Confusing a list of distinct prime divisors with the full factorization including exponents.
Frequently asked questions
What is the prime factorization of 72?
72 = 2³ × 3².
Can a prime number be factorized further?
Not into smaller positive prime factors. Its prime factorization is the number itself.
Does factor order matter?
No. Multiplication is commutative; 2 × 3 × 5 and 5 × 2 × 3 represent the same product.
Why is prime factorization useful?
It helps find common factors and multiples, simplify fractions, and understand divisibility.
Try the Prime Factorization Calculator to check your work.
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