How to use the Prime Factorization
- Enter a whole number greater than one.
- Read the factorisation in exponent notation.
- Check the primality flag if that is all you need.
- Use the factors to simplify radicals or find divisors.
How the calculation works
The fundamental theorem of arithmetic guarantees that every integer above one has exactly one prime factorisation, ignoring the order of the factors. That uniqueness is what makes primes the atoms of number theory and why factorisation answers so many other questions at once.
Trial division is used here: test each candidate divisor from 2 upward, dividing out every time it fits, and stop once the candidate exceeds the square root of what remains. Any factor left at that point must itself be prime, because a composite remainder would have had a factor below its own square root. The square-root cutoff is what keeps the method fast for ordinary numbers.
Once you have the factorisation, the divisor count follows immediately: add one to each exponent and multiply. 360 = 2³ × 3² × 5 has (3+1)(2+1)(1+1) = 24 divisors. The same data gives you the GCF and LCM with any other number, and lets you simplify radicals by pulling out squared primes.
Trial division becomes impractical for very large semiprimes — that difficulty is precisely what RSA encryption relies on.
n = p₁^a₁ × p₂^a₂ × … × pₖ^aₖ ; divisor count = (a₁+1)(a₂+1)…(aₖ+1)Source: Fundamental theorem of arithmetic; trial division with square-root bound.
Worked example
Factorising 360 to find how many divisors it has.
- Divide by 2 three times: 360 → 180 → 90 → 45.
- Divide by 3 twice: 45 → 15 → 5.
- 5 is prime, so 360 = 2³ × 3² × 5¹.
- Divisors: 4 × 3 × 2.
360 = 2³ × 3² × 5, with 24 divisors — which is why it is so common in timekeeping and geometry.
Frequently asked questions
Is 1 a prime number?+
No. Primes have exactly two distinct divisors; 1 has only one, and excluding it keeps factorisation unique.
Why stop at the square root?+
Because any composite number must have a factor at or below its square root; anything remaining above it is prime.
How do I use this to simplify a radical?+
Pull out pairs of identical primes: √360 = √(2² × 90) = 6√10.
Why is factoring large numbers hard?+
No efficient classical algorithm is known, which is the basis of RSA public-key cryptography.
Last reviewed September 1, 2026. We review this page whenever the underlying formula, tax year, published rate or standard changes.