Prime Factorisation Calculator
This prime factorisation calculator breaks any whole number down into the unique set of prime numbers that multiply together to make it, the building block behind simplifying fractions and finding the greatest common factor and lowest common multiple. You enter a whole number from 2 up to a billion, and the calculator instantly returns the full prime factorisation written in index form, such as 360 = 2 cubed times 3 squared times 5, alongside two extra figures: the total count of prime factors with multiplicity, meaning repeated primes counted each time they appear, and the number of distinct primes involved. The result updates as you type, so you can try different numbers and watch the factorisation change. The calculator works by trial division, repeatedly dividing out the smallest prime that fits until only 1 remains, which is fast and reliable up to a billion; much larger numbers need specialised algorithms instead. Because of the fundamental theorem of arithmetic, every whole number greater than 1 has exactly one prime factorisation, so the answer is not an estimate but the single correct decomposition, order of factors aside. Use it to check homework, verify a factor tree by hand, simplify a fraction to its lowest terms, or feed the distinct primes and their powers into a related greatest common factor or lowest common multiple calculation.
The formula
By the fundamental theorem of arithmetic, every integer n greater than 1 is a unique product of primes, n = p1^a1 times p2^a2 and so on. The calculator divides out the smallest prime repeatedly, then the next, until what remains is 1.
Worked example
360 divides by 2 three times to give 45, then by 3 twice to give 5, which is prime. So 360 = 2^3 × 3^2 × 5, six prime factors in total across three distinct primes. Enter 360 to confirm.
Frequently asked questions
What is a prime factorisation?
The unique set of prime numbers, with their powers, that multiply to give the number, for example 12 = 2 squared times 3.
Is the factorisation always unique?
Yes. The fundamental theorem of arithmetic says every integer above 1 has exactly one prime factorisation, apart from the order of the factors.
Why is the upper limit a billion?
The calculator uses trial division, which stays fast up to a billion. Much larger numbers need specialised algorithms.
Who this calculator is for
This calculator is for students, programmers and anyone exploring number theory.
What this calculator assumes
- You enter a whole number within the stated range.
- The exact integer algorithm is used.
- Very large results are shown grouped for readability.
Formula and sources
Related calculators
- Prime Factorization Calculator: break a number into prime factors.
- LCM Calculator: lowest common multiple.
- GCF Calculator: greatest common factor.