Prime Factorization

What is prime factorization?

Any non-prime integer can be expressed as a product of prime numbers.

For example, 40 is not a prime number.

Its prime factorization is:

$$ 40 = 2 \cdot 2 \cdot 2 \cdot 5 = 2^3 \cdot 5 $$

Note: If a number (other than zero or one) cannot be broken down into prime factors, it means its only divisors are 1 and itself. In other words, it's a prime number. For example, 7 is only divisible by 1 and 7: $$ 7 = 7 \cdot 1 $$ So, 7 is a prime number.

How to Find the Prime Factorization of a Number

To break a number down into its prime factors, start by dividing it by the smallest prime numbers (2, 3, 5, ...), continuing until only 1 remains.

$$ 2 \ , \ 3 \ , \ 5 \ , \ 7 \ , \ 11 \ , \ 13 \ , ... $$

If a number ends in 0, you can immediately divide it by 2·5.

Example

Let's factor 48 into primes.

Start by writing 48 in a column:

$$ \begin{array}{c|c} b & d \\ \hline 48 & \\ & \\ & \\ & \\ & \\ & \end{array} $$

Since 48 is even, we divide by 2:

$$ \begin{array}{c|c} b & d \\ \hline 48 & 2 \\ 24& \\ & \\ & \\ & \\ & \end{array} $$

24 is also even, so we divide by 2 again:

$$ \begin{array}{c|c} b & d \\ \hline 48 & 2 \\ 24 & 2 \\ 12 & \\ & \\ & \\ & \end{array} $$

12 is even, so we divide by 2 once more:

$$ \begin{array}{c|c} b & d \\ \hline 48 & 2 \\ 24 & 2 \\ 12 & 2 \\ 6 & \\ & \\ & \end{array} $$

6 is even, so we divide by 2 again:

$$ \begin{array}{c|c} b & d \\ \hline 48 & 2 \\ 24 & 2 \\ 12 & 2 \\ 6 & 2 \\ 3 & \\ & \end{array} $$

3 is not divisible by 2, but it is divisible by 3:

$$ \begin{array}{c|c} b & d \\ \hline 48 & 2 \\ 24 & 2 \\ 12 & 2 \\ 6 & 2 \\ 3 & 3 \\ 1 & \end{array} $$

Now that we’ve reached 1, the process is complete.

So, the prime factorization of 48 is:

$$ 48 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 = 2^4 \cdot 3 $$

Example 2

Now, let's factor 60 into primes.

Start by writing 60 in a column:

$$ \begin{array}{c|c} b & d \\ \hline 60 & \\ & \\ & \\ & \\ & \\ & \end{array} $$

Since 60 ends in 0, it's divisible by 2·5:

$$ \begin{array}{c|c} b & d \\ \hline 60 & 2 \cdot 5 \\ 6 & \\ & \\ & \\ & \\ & \end{array} $$

6 is even, so we divide by 2:

$$ \begin{array}{c|c} b & d \\ \hline 60 & 2 \cdot 5 \\ 6 & 2 \\ 3 & \\ & \\ & \\ & \end{array} $$

3 is divisible by 3:

$$ \begin{array}{c|c} b & d \\ \hline 60 & 2 \cdot 5 \\ 6 & 2 \\ 3 & 3 \\ 1 & \\ & \\ & \end{array} $$

Now that we’ve reached 1, the process is complete.

So, the prime factorization of 60 is:

$$ 60 = 2 \cdot 2 \cdot 3 \cdot 5 = 2^2 \cdot 3 \cdot 5 $$

Example 3

Now, let's factor 28.

Start by writing 28 in a column:

$$ \begin{array}{c|c} b & d \\ \hline 28 & \\ & \\ & \\ & \\ & \\ & \end{array} $$

Since 28 is even, we divide by 2:

$$ \begin{array}{c|c} b & d \\ \hline 28 & 2 \\ 14 & \\ & \\ & \\ & \\ & \end{array} $$

14 is even, so we divide by 2 again:

$$ \begin{array}{c|c} b & d \\ \hline 28 & 2 \\ 14 & 2 \\ 7 & \\ & \\ & \\ & \end{array} $$

7 is a prime number, so we divide by 7:

$$ \begin{array}{c|c} b & d \\ \hline 28 & 2 \\ 14 & 2 \\ 7 & 7 \\ 1 & \\ & \\ & \end{array} $$

The process is complete.

So, the prime factorization of 28 is:

$$ 28 = 2 \cdot 2 \cdot 7 = 2^2 \cdot 7 $$

And so on.