Prime Factorization Calculator

What this calculator does

So, you want to break a number down into the primes that multiply together to make it. That is prime factorization, and this tool does it. Enter a whole number bigger than 1, and it returns the number written as a product of prime numbers.

If the number you enter is itself prime, the tool tells you so, since a prime cannot be broken down any further.

How to use it

1. Enter a whole number greater than 1.
2. Press Calculate.

What prime factorization is

A prime number is a whole number greater than 1 whose only divisors are 1 and itself, like 2, 3, 5, and 7. Primes are the building blocks of all the other whole numbers, because every number that is not prime can be built by multiplying primes together. Prime factorization is the act of finding those building blocks: writing a number as a product of primes. So 12 becomes 2 times 2 times 3, and 60 becomes 2 times 2 times 3 times 5. The tool shows the answer as this string of primes joined by multiplication signs.

Every prime, as many times as it divides

The important thing about prime factorization is that repetition counts. If a prime divides the number more than once, it appears more than once in the answer. The factorization of 12 is 2 times 2 times 3, not just 2 times 3, because there are two factors of 2 hidden inside 12. The tool builds this by dividing out the smallest prime it can, over and over, keeping each one it uses, until nothing but 1 remains. That is why you see every copy of every prime, which is exactly what makes the product multiply back to your original number.

Why the factorization is unique

There is a deep fact underneath this, called the fundamental theorem of arithmetic. It says that every whole number greater than 1 has exactly one prime factorization, apart from the order you write the factors in. No matter how you go about breaking 12 down, you will always end up with two 2s and one 3, never anything else. This uniqueness is what makes prime factorization so powerful: it is like a fingerprint for a number, a single definitive way to describe what it is built from.

A worked example

Enter 60. The tool divides by 2 to get 30, by 2 again to get 15, then by 3 to get 5, which is prime. Collecting the primes it used, the factorization is 2 times 2 times 3 times 5. Enter 7 instead, and since 7 is prime and cannot be split, the tool reports that 7 is a prime number.

How it differs from prime factors

It is worth being clear about a subtle difference. This tool gives the full factorization, with primes repeated as often as they divide: 12 as 2 times 2 times 3. A closely related tool, the prime factor calculator, instead lists only the distinct primes that divide a number, without repeats: 12 as just 2 and 3. One tells you the exact product that rebuilds the number; the other tells you which primes are involved. Use this one when you need the complete breakdown, and the prime factor calculator when you only need the set of primes.

Questions people ask

What is prime factorization?

Writing a number as a product of prime numbers. For example, 60 is 2 times 2 times 3 times 5.

Why are some primes repeated?

Because a prime can divide a number more than once. The 12 contains two factors of 2, so its factorization is 2 times 2 times 3.

Is there only one prime factorization?

Yes. By the fundamental theorem of arithmetic, every number above 1 has exactly one, apart from the order of the factors.

What if my number is prime?

Then it cannot be broken down, and the tool tells you it is a prime number.

How is this different from the prime factor calculator?

This gives the full product with repeats, like 2 times 2 times 3. The prime factor calculator lists only the distinct primes, like 2 and 3.

References

On prime factorization. Every integer above 1 is a product of primes, uniquely, as stated by the fundamental theorem of arithmetic.

1. Eric W. Weisstein, "Prime Factorization," from MathWorld, a Wolfram resource, on expressing a number as a product of its prime factors.
2. Eric W. Weisstein, "Fundamental Theorem of Arithmetic," from MathWorld, a Wolfram resource, on the uniqueness of the prime factorization.