This prime factorization calculator breaks any whole number into its prime factors — showing the expanded product, the exponent form, and every factor of the number.
Why Prime Factorization Matters
The Fundamental Theorem of Arithmetic guarantees that every whole number above 1 decomposes into primes in exactly one way — a fingerprint no two numbers share. That uniqueness quietly powers simplifying fractions, finding GCFs and LCMs (feed these exponents into our GCF Calculator), rationalizing radicals, and — at astronomical scales — RSA encryption, whose security is precisely that multiplying two huge primes is easy while un-multiplying the product is (currently) infeasible. Part of our Math Calculators collection.
Worked Example (Verified)
360, the champion example:
- Factorization: 360 = 2 × 2 × 2 × 3 × 3 × 5 = 2³ × 3² × 5
- Factor count from exponents: (3+1)(2+1)(1+1) = 24 factors — and the full list in the results (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360) really has 24 entries ✔ — 360 is a "highly composite" number, which is why ancient astronomers built the 360° circle on it
- How the machine got there: trial division — divide by 2 until odd, then 3, then 5… stopping at √n, since any larger factor would need a smaller partner already tested
Shortcuts the Calculator Uses (and You Can Too)
- Divisibility tells: ends 0/2/4/6/8 → factor 2; digits sum to a multiple of 3 → factor 3; ends 0/5 → 5; digits sum to a multiple of 9 → 9 divides it (so 3 does, twice).
- Stopping rule: once the trial divisor exceeds √n, what remains (if anything) is prime — for 360 the search ends at 19, not 360.
- Exponent form unlocks counting tricks: number of factors = product of (exponent + 1), as the example shows — try it on 2³×3²×5: (4)(3)(2) = 24.
Related Tools
- GCF Calculator — greatest common factors and LCM from these building blocks
- Factorial Calculator — a different kind of "factor"
- Math Calculators — the full collection
Frequently Asked Questions
What is prime factorization?
Writing a number as the product of prime numbers only: 360 = 2³ × 3² × 5. Every integer greater than 1 has exactly one prime factorization (up to order).
How do I find the prime factorization of a number?
Repeatedly divide by the smallest prime that divides evenly (2, then 3, then 5, 7…) until the quotient is prime; collect the divisors. Stop trying new primes once you pass the square root of what remains.
Is 1 a prime number?
No — by definition primes have exactly two distinct factors (1 and themselves); 1 has only one. Excluding it is what makes prime factorization unique (otherwise 6 = 2×3 = 1×2×3 = 1×1×2×3…).
How many factors does a number have?
Add 1 to each exponent in its prime factorization and multiply: 360 = 2³×3²×5¹ → (3+1)(2+1)(1+1) = 24 factors.
What is the prime factorization of 100?
100 = 2² × 5² — a satisfying perfect square of primes, with (2+1)(2+1) = 9 factors: 1, 2, 4, 5, 10, 20, 25, 50, 100.
Is this prime factorization calculator free?
Yes — free, instant, no sign-up.