Numbers: Integers
Prime factorization
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.
To find the prime factorization, we first try to divide #539# 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 once again divide 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, #539=7 \times 7 \times 11#.
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