Prime Factorization

🏆Practice decomposition into prime numbers

Prime Factorization

Prime factorization (or prime decomposition) consists of breaking down a certain number into prime numbers, called factors, whose product (multiplication) results in the original number.

The first method - Factor tree method

Let's take the number we want to factorize and draw 22 branches from it.
We will ask ourselves, which 22 numbers can we find whose multiplication results in this same number, except for the original number and 11.
Let's see if the numbers we found are prime or composite, we will break down the composite ones into two branches again.
We will continue breaking down all the composite numbers until we only have primes, which we will mark with a circle.

The second method - Split Window Method

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 manner until we reach the number 11 and finish the exercise.
All the prime numbers will appear on the right side of the dividing line.

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Write all the factors of the following number: \( 6 \)

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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!


What is factor decomposition (or factorization)?

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.


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The first method - Factor Tree Method

We will take the number we want to factorize and draw 22 branches from it.
For example:
6464

1 - Factor tree method

We will ask ourselves, which 22 numbers can we find whose multiplication results in this same number, except for the original number and 11.
In this example, we will choose the numbers 44 and 1616.
Note that, we could have chosen any pair of numbers whose product was 6464 and, anyway, we would have arrived at the same result. 

We will write the 44 and the 1616 under the branches as follows:

We will write the 4 and the 16 under the branches

Now we will find out if 44 and 1616 are composite numbers. Again, we will look for 22 numbers whose product is 44 or 1616.
Attention! This is valid except for the number itself and 11.
For the number 44, we can find 22 and 22
For the number 1616, we can find 44 and 44

We will note down 2, 24, and 4 below

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

Now we will continue asking
Is 22 a prime number?
Yes.
Is 44 a prime number?
No. We can continue factoring out 22 and 22.
Therefore, we will continue with the tree on the right side and circle all the factors that are prime numbers.

we will circle all the factors that are prime numbers

What do we get?
If we break down 6464 into its prime factors, we will find that:
2×2×2×2×2×2=642 × 2 × 2 × 2 × 2 × 2=64
are the prime factors of 6464.


Another example

Let's take the number 4848 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 4848 as the product of 22 and 2424. On the left, we will break down 4848 as the product of 66 and 88.
We will continue according to the operation steps
On the right, we will only break down 2424. 22 is prime, therefore, we will mark it with a circle.
On the left, we will break down 66 and also 88.
We will continue breaking down all composite numbers until we only have primes, which we will mark with a circle.

A5 - We will continue breaking down all composite numbers until we only have primes


Do you know what the answer is?

The second method: Split-window division method

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 11 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 6060 into prime factors:
We will ask ourselves, what is the smallest prime number by which we can divide it? The answer is 22. 22 is a prime number and 6060 is divisible by it.
We will write 22 next to the number 6060 to the right of the division line.
Let's calculate
60:2=3060:2=30
We will write the quotient (result of the division) below the original number, as follows:

We will write the quotient (result of the division)

Now, we will continue questioning: What is the smallest prime number that divides 3030?
The answer is 22.
We will write 22 on the right again and note the quotient on the left:
30:2=1530:2=15

What is the smallest prime number that divides 30

Now, we will continue asking: What is the smallest prime number that divides 1515?
The answer is 33. We will write it to the right and the quotient to the left.
15:3=515:3=5

15 - What is the smallest prime number that divides 15

Now, we will continue questioning: What is the smallest prime number that divides 55?
The answer is 55.
We will note it on the right and the result on the left.
5:5=15:5=1
We have reached the number 11, 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=2×2×3×560=2 × 2 × 3 × 5

60=2 × 2 × 3 × 5


Examples and exercises with solutions for prime factorization

Exercise #1

Write all the factors of the following number: 6 6

Video Solution

Answer

2,3 2,3

Exercise #2

Write all the factors of the following number: 7 7

Video Solution

Answer

No prime factors

Exercise #3

Write all the factors of the following number: 5 5

Video Solution

Answer

No prime factors

Exercise #4

Write all the factors of the following number: 9 9

Video Solution

Answer

3,3 3,3

Exercise #5

Write all the factors of the following number: 8 8

Video Solution

Answer

2,2,2 2,2,2

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