Prime Factorization Calculator
Find the prime factorization of any positive integer. See also GCF Calculator and LCM Calculator.
How to Find Prime Factorization
Prime factorization breaks a number down into a product of prime numbers. Start by dividing the number by the smallest prime (2), and continue dividing by primes until the result is 1. Each prime used is a factor, and the number of times it divides evenly is its exponent. Every integer greater than 1 has a unique prime factorization (Fundamental Theorem of Arithmetic).
Prime Factorization Method
1. Start with the smallest prime p = 2
2. While n is divisible by p, divide: n = n / p
3. Move to next prime and repeat
4. Continue until n = 1
n = p1^a1 × p2^a2 × ... × pk^ak
Total divisors = (a1+1)(a2+1)...(ak+1)
Example
Prime factorization of 360:
360 ÷ 2 = 180
180 ÷ 2 = 90
90 ÷ 2 = 45
45 ÷ 3 = 15
15 ÷ 3 = 5
5 is prime
360 = 2³ × 3² × 5
Total divisors: (3+1)(2+1)(1+1) = 24
Frequently Asked Questions
What is a prime number?
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. The first few primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29.
Is 1 a prime number?
No. By convention, 1 is not considered a prime number. This ensures the uniqueness of prime factorization (Fundamental Theorem of Arithmetic).
How is prime factorization used?
Prime factorization is used to find GCF and LCM, simplify fractions, solve problems in cryptography (RSA encryption), and analyze number properties.
Is prime factorization unique?
Yes. The Fundamental Theorem of Arithmetic states that every integer greater than 1 has a unique prime factorization, up to the order of the factors.
Solved Examples — Prime Factorization
Example: Find the prime factorization of 504
Solution:
Step 1: 504 ÷ 2 = 252, 252 ÷ 2 = 126, 126 ÷ 2 = 63
Step 2: 63 ÷ 3 = 21, 21 ÷ 3 = 7
Step 3: 7 is prime
Step 4: 504 = 2³ × 3² × 7
Answer: 504 = 2³ × 3² × 7
Example: How many divisors does 180 have?
Solution:
Step 1: Prime factorization: 180 = 2² × 3² × 5¹
Step 2: Number of divisors = (2+1)(2+1)(1+1)
Step 3: = 3 × 3 × 2 = 18
Answer: 18 divisors
Example: Find GCF(120, 84) using prime factorization
Solution:
Step 1: 120 = 2³ × 3 × 5
Step 2: 84 = 2² × 3 × 7
Step 3: Common primes with minimum powers: 2² × 3¹
Step 4: GCF = 4 × 3 = 12
Answer: GCF(120, 84) = 12
Practice Questions
Try these on your own:
- Find the prime factorization of 252 (Answer: 2² × 3² × 7)
- How many divisors does 72 have? (Answer: 12)
- Is 97 a prime number? (Answer: Yes)
- Find the prime factorization of 1000 (Answer: 2³ × 5³)
- Using prime factorization, find LCM(24, 36) (Answer: 72)
- What is the sum of prime factors of 330? (Answer: 2 + 3 + 5 + 11 = 21)
Common Mistakes to Avoid
A common mistake is stopping the factorization too early — always continue dividing until the remaining number is 1. For instance, 28 = 2 × 14 is NOT complete; it should be 28 = 2² × 7. Another error is including 1 as a prime factor — 1 is not prime by definition. Students sometimes skip testing primes and incorrectly include composite numbers (like 4, 6, 9) as factors. You only need to test primes up to √n to determine if n itself is prime. Also, remember that the same prime can appear multiple times — in 8 = 2³, the factor 2 appears three times. When using factorization to find GCF, use the MINIMUM powers, and for LCM, use the MAXIMUM powers.
Key Takeaways
- Every integer > 1 has a unique prime factorization (Fundamental Theorem of Arithmetic).
- Method: divide by smallest prime (2, 3, 5, 7, ...) repeatedly until quotient = 1.
- Number of divisors = product of (exponent + 1) for each prime factor.
- To check if n is prime, test divisibility by all primes up to √n.
- Prime factorization is the foundation for finding GCF and LCM.
- Applications: cryptography (RSA uses large primes), simplifying fractions, and number theory.