Prime factorization calculator

Find the prime factorization of any number and check if it is prime

Every whole number from 2 upwards can be written as a product of primes in exactly one way, apart from the order of the factors: this is the fundamental theorem of arithmetic. 360 = 2³ × 3² × 5, 1,001 = 7 × 11 × 13, and 97 cannot be split at all: it is prime. This calculator gives the factorization, tells you instantly whether the number is prime or composite, and shows the division ladder exactly as it is taught in class.

Prime factorization unlocks many everyday calculations: simplifying fractions, finding a GCD or LCM, counting divisors, simplifying square roots. It is used by middle and high school students, teachers preparing worksheets, math competition entrants and anyone curious about cryptography: the security of RSA relies on how hard it is to factor very large numbers.

1. Type the number

   Enter a whole number in “Whole number to factor” (at least 2, up to 18 digits). Spaces and thousands separators are ignored, so 600,851,475,143 works as is. The “Try:” buttons load ready-made examples.

2. Read the verdict

   The badge says “Prime number” or “Not prime: composite number”, then the factorization appears with exponents (360 = 2³ × 3² × 5) and its expanded form. The “Copy” button puts 2^3 × 3^2 × 5 on your clipboard.

3. Explore the properties

   The “Arithmetic properties” cards give τ(n), σ(n), φ(n), the number of distinct prime factors ω(n) and of prime factors with multiplicity Ω(n), whether n is a perfect square, the sum of proper divisors, and the previous and next primes (up to 10¹²).

4. Check it by hand

   The “Division ladder” table repeats every step (360 ÷ 2 = 180…) down to quotient 1, and “All divisors” lists every divisor in increasing order, with the prime divisors highlighted.

The method: fundamental theorem and formulas

For a whole number n ≥ 2, the factorization is written n = p1a1 × p2a2 × … × pkak. You get it by dividing by the smallest prime factor for as long as possible; you only need to test divisors up to √n, because a composite number always has a prime factor no larger than its square root.

For large inputs the tool chains three stages: trial division by the primes up to 10,000, a deterministic Miller-Rabin test to recognize primes, then Pollard’s rho algorithm (Brent’s variant) to split the remaining composite cofactors. Every calculation uses exact integers, with no rounding.

• p₁ < p₂ < … < p_k: the distinct prime factors in increasing order; how many there are is written ω(n).
• a_i: the exponent (multiplicity) of p_i; the sum a₁ + … + a_k is Ω(n), the number of prime factors counted with repetition.
• τ(n) = (a₁ + 1)(a₂ + 1)…(a_k + 1): the number of divisors, since a divisor picks an exponent from 0 to a_i for each p_i.
• σ(n) = ∏ (p_i^a_i+1 − 1) / (p_i − 1): the sum of all divisors; s(n) = σ(n) − n tells whether the number is perfect, abundant or deficient.
• φ(n) = n × ∏ (1 − 1/p_i): Euler’s totient, the count of integers from 1 to n that are coprime to n.
• Perfect square: n is a square exactly when every a_i is even; then √n = ∏ p_i^a_i/2, for example √1,024 = 2⁵ = 32.

Prime factorization examples

What is prime factorization used for?

• Simplifying fractions: 84/360 = (2² × 3 × 7)/(2³ × 3² × 5) = 7/30 once the common factors 2² × 3 = 12 are cancelled.
• Finding a common denominator: the LCM of 360 and 84 is 2,520, the least common denominator for adding 1/360 and 1/84.
• Simplifying square roots: √360 = √(2² × 3² × 10) = 2 × 3 × √10 = 6√10: each pair of equal factors comes out of the root.
• Counting and listing divisors: handy for grouping problems (arranging 360 chairs in equal rows: 24 possible rows × columns layouts) and number theory homework.
• Cryptography and puzzles: RSA relies on factoring being hard; the famous Project Euler problem 3 asks for the largest prime factor of 600,851,475,143, which is 6,857.

Common mistakes

• Dividing by a non-prime: dividing 360 by 4 or 6 gives a correct quotient but an incomplete factorization; only divide by 2, 3, 5, 7, 11, 13…
• Treating 1 as prime: 1 has a single divisor and never appears in a factorization.
• Mixing up 2³ and 2 × 3: 2³ = 2 × 2 × 2 = 8, whereas 2 × 3 = 6; the exponent counts how many times the factor repeats.
• Assuming a number “looks prime”: 51 = 3 × 17, 91 = 7 × 13 and 1,001 = 7 × 11 × 13 are all composite; always test up to the square root.

All calculations run locally in your browser with exact integer arithmetic (BigInt): no number is sent or stored. Limits: 18 digits maximum; neighboring primes are shown up to 10¹², the divisor list up to 1,000 divisors and the division ladder up to 40 prime factors counted with multiplicity.