Prime Factorization Calculator

Every integer greater than 1 can be written uniquely as a product of prime numbers — this is the Fundamental Theorem of Arithmetic. Prime factorization breaks a composite number into its prime building blocks, typically displayed in exponential form. The number 360 decomposes as 360 = 2³ × 3² × 5 because 360 = 8 × 9 × 5 = 2×2×2 × 3×3 × 5. The exponents show how many times each prime appears.

The trial division algorithm tests small primes in order: divide by 2 while possible, then 3, then test odd candidates 5, 7, 11, … up to √remaining. Each successful division is recorded as a step. For 100: 100 ÷ 2 = 50, 50 ÷ 2 = 25, 25 ÷ 5 = 5, 5 ÷ 5 = 1 — yielding 100 = 2² × 5². A number is prime if the factorization contains only itself: 97 = 97 with no smaller divisors.

Factor trees offer a visual alternative taught in schools: repeatedly split composite nodes into smaller factors until all leaves are prime. The tree for 360 might branch 360 → 36 × 10 → (6×6) × (2×5) → … arriving at the same 2³ × 3² × 5 regardless of branch choices. This calculator shows sequential division steps equivalent to a factor tree's logic.

Prime factorization powers GCF and LCM computation: GCF takes the minimum exponent of each shared prime; LCM takes the maximum. It simplifies radicals (√72 = √(2³×3²) = 6√2), counts divisors (add 1 to each exponent and multiply: (3+1)(2+1)(1+1) = 24 divisors for 360), and underpins cryptographic key generation where enormous primes are multiplied together. Link to the factor calculator to list all divisors from the prime exponents.

Special cases: 1 has no prime factors (empty product by convention). 2 is the only even prime. All other primes are odd. Large composites like 360 = 2³×3²×5 and 100 = 2²×5² illustrate how the same primes recur across numbers — shared factors drive the GCF calculator and fraction simplification.

Enter any positive integer and receive exponential form, the list of distinct primes, step-by-step division, and a primality flag. Use results to verify homework, prepare for algebra lessons on radicals and rationals, or explore number theory patterns in composite and prime structure.

Divisors count formula from exponents: if n = p₁^a × p₂^b × …, the number of positive divisors is (a+1)(b+1)…. For 360 = 2³×3²×5, divisor count = (3+1)(2+1)(1+1) = 24. Cross-check with the factor calculator to list all 24 divisors explicitly and confirm the exponential decomposition is correct.