## 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.

Examples:

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:

12 |

2 |

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.

Example:

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: 2^{3} · 3 · 5^{2}

## 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.