Prime Factorization Calculator
Find the complete prime factorization of any number with step-by-step division. Shows exponential notation, all factors, and GCF/LCM applications.
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How It Works
Divide repeatedly by the smallest prime (2) until odd. Then try 3, 5, 7... up to √n. Each successful division gives a prime factor. The final quotient > 1 is also prime.
Divide by primes in order until quotient = 1360÷2=180÷2=90÷2=45÷3=15÷3=5 → 2³×3²×5Group repeated prime factors using exponents. 360 = 2×2×2×3×3×5 written compactly as 2³×3²×5. This form makes GCF, LCM, and factor counting much easier.
n = p1^a1 × p2^a2 × ...360 = 2³ × 3² × 5The number of factors of n = p1^a1 × p2^a2... is (a1+1)(a2+1)... Multiply (each exponent + 1). For 360 = 2³×3²×5¹: (3+1)(2+1)(1+1) = 24 factors.
(a1+1)(a2+1)(a3+1)...360: (3+1)(2+1)(1+1)=24 factorsFor GCF: write factorization of each number. Find primes common to all. Take the lowest exponent of each common prime. Multiply these together.
GCF = product of common primes at min powerGCF(24,36): 24=2³×3, 36=2²×3² → GCF=2²×3=12For LCM: take all primes that appear in any factorization. Use the highest exponent of each. Multiply together.
LCM = product of all primes at max powerLCM(24,36): 24=2³×3, 36=2²×3² → LCM=2³×3²=72RSA encryption relies on the fact that multiplying two large primes is fast, but factoring their product is computationally hard. A 2048-bit RSA key involves primes with ~300 digits.
Security: easy to multiply, hard to factor15 = 3×5 easy; 5561 = 67×83 requires workQuick Reference
Common examples — verify instantly above.
Simple
12 factorized
2² × 3
Simple
60 factorized
2² × 3 × 5
Power of 2
64 factorized
2⁶
Three primes
210 factorized
2 × 3 × 5 × 7
Square
900 factorized
2² × 3² × 5²
Large
360 factorized
2³ × 3² × 5
Prime
97 factorized
97 (prime)
Factors
Factors of 360
24 total
Tips & Shortcuts
Always start with 2. Divide by 2 until odd, then move to 3, then 5, 7... You never need to test composite divisors.
You only need to test primes up to √n. If none divides n, then n is prime.
The number of factors = product of (each exponent + 1). Quick check: 12 = 2²×3 → (2+1)(1+1) = 6 factors.
For GCF: use lowest exponents of shared primes. For LCM: use highest exponents of all primes.
360 = 2³×3²×5 is a useful benchmark — it has 24 factors and divides many common numbers.
A perfect square has only even exponents in its prime factorization. 36 = 2²×3² — both exponents are even.
Common Mistakes
Stopping factorization early: writing 4×3 instead of 2²×3 for 12
Continue factoring until all factors are prime. 4 is not prime — it factors to 2×2.
Including 1 as a prime factor
1 is not prime and not included in prime factorization. Every number has 1 as a trivial factor, but it carries no information.
Thinking large numbers have many prime factors
Primes and near-primes have few or one prime factor. 997 is prime — its factorization is just 997.
Using GCF logic (lowest powers) when computing LCM
GCF uses lowest powers of common primes. LCM uses highest powers of ALL primes. They use opposite selection rules.
Confusing factor count with prime factor count
360 has 24 total factors but only 3 distinct prime factors (2, 3, 5). Factor count formula uses exponents, not prime count.
Applying prime factorization to 1
1 is the only positive integer with no prime factorization. The empty product equals 1.
Frequently Asked Questions
Prime factorization expresses a number as a product of prime numbers. Every integer greater than 1 has a unique prime factorization. Example: 360 = 2³ × 3² × 5.
The Fundamental Theorem of Arithmetic states every integer > 1 has exactly one prime factorization (ignoring order). This uniqueness is foundational to number theory.
Divide by the smallest prime (2) as many times as possible. Move to next prime (3), repeat. Continue until the quotient is 1. All divisors used are the prime factors.
A factor tree splits a number into two factors, then splits each factor further until all branches end in primes. The primes at the branch tips are the prime factors.
1 is not prime. It has only one factor (itself). The Fundamental Theorem of Arithmetic requires 1 to be excluded so that prime factorization is unique.
GCF takes the lowest power of each common prime. LCM takes the highest power of each prime across all numbers.
RSA encryption — the system securing HTTPS websites — relies on the fact that multiplying two large primes is trivially fast, but factoring their product back into the original primes is computationally infeasible. A 2048-bit RSA key uses primes each about 300 digits long. Their product would take longer than the age of the universe to factor with current algorithms, making communications secure.
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