Prime Factorization Calculator
Free online prime factorization calculator. Find prime factors, factor tree, exponent form, and all divisors of any number up to 10 million.
Prime factorization breaks a composite number into its prime building blocks. Every integer greater than 1 either is prime itself or can be expressed as a unique product of prime numbers (the Fundamental Theorem of Arithmetic). This tool finds all prime factors and shows them in exponent form.
How to Find Prime Factors
- Trial division: Divide repeatedly by 2, then 3, 5, 7, 11 … until the quotient is 1
- Example — 360: 360 = 2³ × 3² × 5 = 8 × 9 × 5
- Tip: A number is prime if it has no divisors up to its square root
- Divisor count: For n = p₁ᵃ × p₂ᵇ × …, the count of all divisors is (a+1)(b+1)…
Uses of Prime Factorization
Finding GCD (take the minimum exponent of each shared prime) and LCM (take the maximum exponent) both rely on prime factorization. In cryptography, the difficulty of factoring very large numbers into primes is the mathematical foundation of RSA encryption that secures internet communication.
Worked example: factor 360 into primes
Divide by the smallest prime that divides exactly: 360 ÷ 2 = 180, 180 ÷ 2 = 90, and 90 ÷ 2 = 45. Then 45 ÷ 3 = 15, 15 ÷ 3 = 5, and 5 ÷ 5 = 1.
Collecting these factors gives 360 = 2³ × 3² × 5. Check the answer by multiplying 8 × 9 × 5 = 360. The number 1 has no prime factors; a prime number has only itself as its prime factor. Use positive whole numbers greater than 1 for this calculation.