Prime Factorisation Calculator

This prime factorisation calculator breaks a whole number down into the prime numbers that multiply together to make it. Enter a number from 2 up to a billion and the calculator returns its prime factorisation in index form, such as 360 = 2 cubed times 3 squared times 5, along with the total number of prime factors counted with repetition and the number of distinct primes. Every integer greater than one has exactly one prime factorisation, a result known as the fundamental theorem of arithmetic, which is why this is a building block for finding factors, simplifying fractions, and working out the greatest common factor and lowest common multiple. The method, a worked example and the assumptions are set out below.

Whole number (2 to 1,000,000,000)

Prime factors (with multiplicity)

Distinct primes

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 \u00D7 3^2 \u00D7 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.

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