Prime Factorization

In this article, you will learn to break down any number into prime factors with the help of two magnificent tricks that you will surely love!

Factor decomposition (factorization or prime factorization) is the decomposition of a certain number into smaller numbers - primes, whose product (multiplication) is the original number.
After having learned what a composite number and a prime number are, we will learn here how to decompose a composite number into prime factors.

We will take the number we want to factorize and draw $2$ branches from it.
For example:
$64$

We will ask ourselves, which $2$ numbers can we find whose multiplication results in this same number, except for the original number and $1$.
In this example, we will choose the numbers $4$ and $16$.
Note that, we could have chosen any pair of numbers whose product was $64$ and, anyway, we would have arrived at the same result. 

We will write the $4$ and the $16$ under the branches as follows:

Now we will find out if $4$ and $16$ are composite numbers. Again, we will look for $2$ numbers whose product is $4$ or $16$.
Attention! This is valid except for the number itself and $1$.
For the number $4$, we can find $2$ and $2$
For the number $16$, we can find $4$ and $4$

Again, we will extract roots from each number and write the factors in the following way:

Now we will continue asking
Is $2$ a prime number?
Yes.
Is $4$ a prime number?
No. We can continue factoring out $2$ and $2$.
Therefore, we will continue with the tree on the right side and circle all the factors that are prime numbers.

What do we get?
If we break down $64$ into its prime factors, we will find that:
$2 × 2 × 2 × 2 × 2 × 2=64$
are the prime factors of $64$.

Let's take the number $48$ and see how we arrive at the same result even if we start breaking it down with different factors:
On the right, we will break down $48$ as the product of $2$ and $24$. On the left, we will break down $48$ as the product of $6$ and $8$.
We will continue according to the operation steps
On the right, we will only break down $24$. $2$ is prime, therefore, we will mark it with a circle.
On the left, we will break down $6$ and also $8$.
We will continue breaking down all composite numbers until we only have primes, which we will mark with a circle.

Let's write the number we want to factorize on the left side of a vertical line that acts as a division window.
Let's look for the smallest prime number by which we can divide the original, we write it on the right side of the line and the result we write on the left, below the first one. We will continue in this way until we reach the number $1$ and finish the exercise.
All the prime numbers will appear on the right side of the dividing line.

Let's see it in practice: We will break down the number $60$ into prime factors:
We will ask ourselves, what is the smallest prime number by which we can divide it? The answer is $2$. $2$ is a prime number and $60$ is divisible by it.
We will write $2$ next to the number $60$ to the right of the division line.
Let's calculate
$60:2=30$
We will write the quotient (result of the division) below the original number, as follows:

Now, we will continue questioning: What is the smallest prime number that divides $30$?
The answer is $2$.
We will write $2$ on the right again and note the quotient on the left:
$30:2=15$

Now, we will continue asking: What is the smallest prime number that divides $15$?
The answer is $3$. We will write it to the right and the quotient to the left.
$15:3=5$

Now, we will continue questioning: What is the smallest prime number that divides $5$?
The answer is $5$.
We will note it on the right and the result on the left.
$5:5=1$
We have reached the number $1$, which cannot be factored further.
That means we have finished the exercise and the prime factors are those that appear on the right side.
That is:
$60=2 × 2 × 3 × 5$