Prime factorization - calculator

About the calculator

This calculator can be used to perform the prime factorization for a number. Optionally, the prime factorization can also be displayed in canonical representation if it contains identical prime factors.

What is a prime number?

A prime number is a natural number that is divisible by exactly 2 positive integers. The two possible divisors are the number 1 and the prime number itself.

For example, the numbers 2, 3, 5, 7, 11, and 13 are prime numbers because they can each be divided by 1 and by themselves without remainder, but by no other natural number.

The number 4, on the other hand, is not only divisible by 1 and 4, but also by 2. Thus, 4 is not a prime number.

Why is 1 not a prime number?

By definition, a prime number is divisible by exactly 2 natural numbers. The number 1, on the other hand, is divisible by only one natural number (namely 1) and thus 1 is not a prime number.

Prime factorization

Every natural number greater than 1 is either a prime number itself or can be represented as a product of prime numbers. The individual factors of this product are called prime factors. A prime factorization is the representation of a natural number as a product of prime numbers.


The prime factorization of 60 looks like this: 2 · 2 · 3 · 5
The prime factors are therefore 2, 2, 3 and 5.

The number 17 is already a prime number and is therefore not split further in the prime factorization.

The prime factorization of 45 looks like this: 3 · 3 · 5

What is the procedure for prime factorization?

Case - the number is a prime number:

If the number for which the prime factorization is to be performed is already a prime number, then the number cannot be factorized any further and you are done.

Case - the number is not a prime number:

If the number is not a prime number, then look for a divisor of the number. Optimally, the divisor is a prime number, but this is not mandatory. Next, replace the number with the product of the divisor and the quotient of the number and the divisor. For example, if you want to calculate the prime factorization of 12 and you have chosen 2 as the divisor, replace 12 by 2 · 6 (because
= 6

If now not all numbers of the product are prime numbers, then determine the divisor for a factor which is not a prime number and replace the factor again by a product where one of the factors is the determined divisor. This is repeated until all factors of the product are prime numbers.


The prime factorization for the number 420 is to be performed.

420 is obviously divisible by 2. So the following applies:

420 = 2 · 210

2 is a prime number and therefore cannot be partitioned further. 210 is not a prime number and is divisible by 2. So you can replace 210 by the product of 2 and 105.

420 = 2 · 2 · 105

The 105 is divisible by 5:

420 = 2 · 2 · 5 · 21

The 21 can still be replaced by 3 · 7:

420 = 2 · 2 · 5 · 3 · 7

Now all factors of the product are prime numbers and thus one is done. If you want, you can sort the factors according to their size for better readability.

420 = 2 · 2 · 3 · 5 · 7

To find out whether a factor is a prime number or to search for suitable divisors, a list of prime numbers can be useful.

canonical prime factorization

In the canonical representation of a prime factorization, equal factors are combined into powers.

For example, the prime factorization of the number 600 is: 2 · 2 · 2 · 3 · 5 · 5

Converted to canonical representation this is: 23 · 3 · 52

What is the prime factorization needed for?

The prime factorization can be used to determine the greatest common divisor (GCD) or the least common multiple (lcm) of a number. The GCD can be used to reduce a fraction to its lowest terms. The lcm can be used to convert 2 fractions into fractions with equal denominators to be able to add or subtract them.

good explanatory videos on YouTube: