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 #171#.

#3 \times 3 \times 19#

To find the prime factorization, we first try to divide #171# 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, #171=3 \times 3 \times 19#.

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