Prime factorization

Numbers: Integers

Prime factorization

Prime factor

We can factorize the number $60$ : $60=2\times 2 \times 3 \times 5$

In this case, all factors are prime numbers.
A factor that is a prime number is called a $\green{\textbf{prime factor}}$ .

A factorization with only $\green{\textbf{prime factors}}$ is also called a $\green{\textbf{prime factorization}}$ of an integer.

The prime factorization of an integer is unique. That means there is only one possible prime factorization.

Examples

$\begin{array}{rcl} 4 &=& \green{2} \times \green{2} \qquad \\ 6 &=& \green{2} \times \green{3} \\ 8 &=& \green{2} \times \green{2} \times \green{2} \\ 9 &=& \green{3} \times \green{3} \\ 10 & =& \green{2}\times \green{5} \\12 & = & \green{3}\times \green{4} \\ 14 & =& \green{2} \times \green{7} \\15 & =& \green{3} \times \green{5} \\ 16 &=& \green{2} \times \green{2} \times \green{2} \times \green{2} \end{array}$

Prime factorization is generally possible.

Prime factorization

Each positive integer that is not a prime number can be written as a prime factorization.

Prime factorization for prime numbers

For prime numbers, we do not have a prime factorization since we defined a factorization to have two factors that are smaller than the integer we started with. However, when working with prime factorizations, we often say that the prime factorization of a prime number is just the prime number itself.

Give the prime factorization of $455$ .

$5 \times 7 \times 13$

To find the prime factorization, we first try to divide $455$ by the smallest prime number, namely $2$ . If the result is an integer, $2$ is part of the prime factorization. In that case, we try to divide once again the result of our division by $2$ .
If we cannot divide by $2$ or cannot divide by $2$ anymore, we do the same with the next prime number, which is $3$ . We will continue this way until we have a factorization consisting only of prime numbers.
In this case, $455=5 \times 7 \times 13$ .

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