1. Divide by the smallest prime
Start with 2, then test larger possible divisors until the remaining quotient is prime.
Prime factors and divisor analysis
Prime Factorization Calculator
Break a positive integer into prime factors, display exponent form and factor pairs, list all divisors, test primality, and calculate divisor count, divisor sum, Euler’s totient, radical, square-free status, GCF, and LCM.
Prime factor tree
Complete factor list
GCF and LCM comparison
Prime factorization
2³ × 3² × 5
Number
360
Divisors
24
Find the prime factorization of an integer
Enter a whole number from 2 to 1,000,000,000,000. Add a second number to calculate GCF and LCM.
Positive divisors
Factor pairs
Division steps
Comparison factorization
840 = 2³ × 3 × 5 × 7
Prime factorization applies to integers greater than 1. The number 1
is neither prime nor composite and has no prime factorization.
How the Prime Factorization Calculator works
The calculator repeatedly divides the number by the smallest available prime factor. It records each prime and exponent, then uses the factorization to derive divisor, totient, radical, GCF, and LCM results.
2. Group repeated prime factors
Repeated factors are written with exponents, such as 2 × 2 × 2 = 2³.
3. Derive number properties
Exponents determine divisor count and sum, while distinct primes determine the radical and Euler’s totient.
Divisor Count
If n = p₁^a₁ ··· pₖ^aₖ, then d(n) = (a₁+1)···(aₖ+1)
Euler’s Totient
φ(n) = n × ∏(1 − 1/p) over distinct prime factors p
Greatest Common Factor and LCM
gcd(a,b) × lcm(a,b) = a × b
Number type
Definition
Example
Prime number
Exactly two positive divisors
13 is prime because its only positive divisors are 1 and 13.
Composite number
More than two positive divisors
12 is composite because 12 = 2² × 3 and has six positive divisors.
Square-free number
No repeated prime factor
30 = 2 × 3 × 5 is square-free, while 12 = 2² × 3 is not.
Perfect square
Every prime exponent is even
144 = 2⁴ × 3² is a perfect square because all exponents are even.
Frequently asked questions
Important details about primes, factors, divisors, GCF, and LCM.
What is prime factorization?
Prime factorization expresses a positive integer greater than 1 as a product of prime numbers.
Is every prime factorization unique?
Yes. The Fundamental Theorem of Arithmetic says every integer greater than 1 has a unique prime factorization, apart from the order of the factors.
Is 1 a prime number?
No. A prime number has exactly two positive divisors, but 1 has only one positive divisor.
How do exponents work in prime factorization?
An exponent shows how many times a prime factor appears. For example, 72 = 2³ × 3².
How is the number of divisors calculated?
Add 1 to every prime exponent and multiply the results. For 72 = 2³ × 3², the count is (3+1)(2+1) = 12.
What is Euler’s totient function?
Euler’s totient φ(n) counts the positive integers up to n that are relatively prime to n.
How are GCF and LCM found from prime factors?
The GCF uses the smallest shared prime exponents. The LCM uses the largest prime exponents appearing in either number.
What does square-free mean?
A square-free number has no prime factor repeated twice or more, so every exponent in its prime factorization is 1.