﻿ Factorization: How Is It Done? Solved exercises step by step

# Factorization: How Is It Done? Solved exercises step by step

Next we are going to explain what it is, what it is for and how factorization is done step by step, with examples resolved step by step.

## What Factorization Is

What is factoring a number?

Factorial decomposition, or factorization of a number into prime factors, consists of expressing a number as the multiplication of its prime factors.

Prime numbers are those that are only divisible between themselves and between 1. Learning prime numbers between 1 and 30 is more than enough. I personally don’t know any more:

2 – 3 – 5 – 7 – 11 – 13 – 17 – 19 – 23 – 29

Therefore, we can express a number as a product of factors, as for example:

12 = 4 x 3

We have expressed 12 as the product of two factors 4 and 3. But this is not a factorization because 4 is not a prime number.

But if we express it in this other way:

12 = 2 x 2 x 3

This is a decomposition into prime factors. 2 and 3, which are the factors in which we now express 12, are prime numbers.

A little further down we will see how factorial decomposition is done

## What is Factorization for?

We use factorial decomposition or factorization into prime factors:

• Calculate the maximum common divisor.
• Calculate the minimum common multiple.
• To simplify fractions.

## How Factorization of a number is done

For greater clarity, let’s look directly with an example at the procedure for breaking down a number into a product of prime factors. Let’s do the factorial decomposition of 6:

1 – Write the number to be factored and draw a line to the right. To the right of this line will appear the prime factors of the number.

2 – It is necessary to look for if the 6 is divisible between some prime number. Now is when the rules of divisibility come into play.

You start by looking if it is divisible with the first prime number, i.e. with 2 (using the divisibility rule of 2).

If the number is not divisible by 2, then it is tested with 3 (using the divisibility rule of 3).

If once again the number is not divisible by 3, then follow with 5 (using the divisibility rule of 5).

You always start by checking if the number is divisible by 2 and then follow in ascending order with the rest of prime numbers: with 3, with 5…

• Is it divisible by 2? We check it with the divisibility rule of 2
• If it is not divisible by 2, is it divisible by 3? We check it with the divisibility rule of 3
• If it is not divisible by 3, is it divisible by 5? We check it with the rule of divisibility of 5
• And so on…

As I commented before, we always start with 2 and for this we use the divisibility rule of 2.

We ask ourselves, is 6 divisible by 2? Yes, because it is an even number.

Well, we write it at the same height as 6, to the right of the line: Now we divide 6 by 2: And the result is placed below 6: We already have the first factor.

3 – Now we repeat the previous step but in this case with the 3, which was the result of doing the division previously.

We start looking if it is divisible by 2:

Is 3 divisible by 2? Not because it doesn’t end in par

As it is not divisible by 2, now we continue with 3:

Is 3 divisible by 3? Yes. Besides, 3 is a prime number so directly the 3 is placed to the right of the line: And we make the division of the last number we have on the left, in this case the 3, with the last factor we have on the right, in this case, another 3: And the result is placed below the last number to the left of the line, that is, below the 3: And we already have the 6 broken down into prime factors.

When the last number we have to the right is a prime number and therefore, the result of the division is 1, it means that we have finished decomposing.

4 – The numbers to the right of the line are the factors. Therefore the 6 can be expressed as:

6 = 2 x 3

Let’s repeat this procedure with another example, breaking down the number 40:

Draw the line and place it to its left: As always, we start with 2. Is 40 divisible by 2? Yes, because it ends in 0. We write it to the right: We make the division and place the result underneath:  Now we have the 20 and we start again.

Is 20 divisible by 2? Yes because it ends in 0. We rewrite the 2 to the right of the line, divide 20 by 2, and place the result underneath:   We have the 10.

We start again again is 10 divisible by 2? Yes, because it ends in 0. We divide and place the result to the left:   We have 5 left, which is a prime number, so we write on the right side, divide and we are done:   The factors are those on the right and the 40 expressed as a product of their prime factors is: If you don’t realize that 5 is a prime number at first, you would realize it later, following the usual process:

• Is 5 divisible by 2? Not because it ends in 5
• Is 5 divisible by 3? Not because the sum of its figures is not a multiple of 3.
• Is 5 divisible by 5? Yes, because it ends in 5. In fact, it can only be divided by 5 and by 1 (5 is prime).

Notice that every time we repeat the process, we always start with the 2 and we do it until we can’t do it anymore. The same would be done later with the 3

We must divide by the same prime number as many times as possible.

Let’s see now other examples of greater difficulty, with the numbers 1225 and 540:  Following this method, you can make the factorial decomposition of 49, 65, 98, 121 and any number.

## Remember

• It is very important to follow the order when testing with prime numbers.
• Apart from the Rules of Divisibility there are other methods to know the divisibility of a number. For example, we know that 49 is divisible by 7 because 49 is the square of 7.