Title: LESSON: Prime Factorization
Course: Mathematics Essentials Q1 > Section: Week 6

Prime Factorization

INTRODUCTION

Prime Factorization

READ: Find Factor Pairs of Given Numbers

Find Factor Pairs of Given Numbers

This lesson is all about factors. That is where we are going to start. In order to complete any of the work in this lesson, you will first need to understand and identify a factor.

What is a factor?

When you multiply, the numbers that are being multiplied together are the factors of the product. Said another way, a factor is the two or more numbers that are multiplied together for a product.

In this lesson, you will be finding factor pairs this is when only two numbers are multiplied together for a product.

Let’s find some factors.

Example

What are two factors for twelve?

Here we want to find two factors of twelve or two numbers that multiply together to give us twelve. We could list many possible factors for twelve. Let’s choose 3 and 4.

Our answer is 3 4.

What if we wanted to list out all of the factors for twelve? To do this systematically, we should first start with the number 1. Yes, one is a factor of twelve. In fact, one is a factor of every number because any number can be multiplied by one to get itself as a product.

1 12

After starting with 1, we can move on to 2 then 3 and so on until we have listed out all of the factors for 12.

5, 7, 8 etc are not factors of 12 because we can’t multiply them by another number to get 12.

These are all of the factors for 12.

READ: Use Divisibility Rules to Find Factors

Use Divisibility Rules to Find Factors of Given Numbers

When we have a larger number that we are factoring, we need to use divisibility rules to help us find the factors of that number.

What are divisibility rules?

Divisibility rules looks at different numbers in a given number. Depending on the numbers or combinations of numbers, we can determine if a number is divisible by let’s say 2 or 3 or 4. This can help us to identify the factors of a number.

Here is a chart that shows all of the divisibility rules.

Characteristic of Number | Number It’s Divisible By | Example:
Last digit is even | 2 | 208
The sum of all the digits is divisible by 3 | 3 | 513 5 + 1 + 3 = 9 9 is divisible by 3, so 513 is also divisible by 3.
The last two digits are divisible by 4 | 4 | 616 16 is divisible by 4, so 616 is also divisible by 4.
The last digit is 0 or 5 | 5 | 590
The number is divisible by 2 and 3 | 6 | 438 Last digit is even and 4 + 3 + 8 = 15, which is divisible by 3.
Double the last digit, subtract it from the rest of the number (not including the last digit). If that number is divisible by 7, the original number is divisible by 7.
7
574

4×2 = 8

57 – 8 = 49

49 is divisible by 7, so 584 is also divisible by 7.
The last 3 digits are divisible by 8 | 8 | 1,856 856 is divisible by 8, so 1,856 is also divisible by 8.
The sum of the digits is divisible by 9 | 9 | 567 5 + 6 + 7 = 18 18 is divisible by 9, so 567 is also divisible by 9.
The last digit is a 0 | 10 | 1,560
The number is divisible by both 3 and 4 | 12 | 1,824 1 + 8 + 2 + 4 = 15, which is divisible by 3. 24 is divisible by 4, so 1,824 is also divisible by 4. Since 1,824 is divisible by 3 and 4, it is also divisible by 12.

Now some of these rules are going to be more useful than others. But you can use this chart to help you.

Example

What numbers is 1346 divisible by?

To solve this, we can go through each rule and see if it applies.

1. The last number is even-this number is divisible by 2.
2. The sum of the digits is 14-this number is not divisible by 3.
3. The last two digits are not divisible by 4-this number is not divisible by 4.
4. The last digit is not a zero or five-this number is not divisible by 5.
5. 6x2=12; 134 - 12 = 122. 122 is not divisible by 7-this number is not divisible by 7.
6. 346 is not divisible by 8-this number is not divisible by 8.
7. The sum of the digits is 14-this number is not divisible by 9
8. The number does not end in zero-this number is not divisible by 10
9. The number is not divisible by 3 and 4.

Our answer is that this number is divisible by 2.

You won’t usually have to go through each rule of disability, but it is important that you know and understand them just in case.

READ: Classify Given Numbers as Prime or Composite

Classify Given Numbers as Prime or Composite

Now that you have learned all about identifying and finding factors, we can move on to organizing numbers. We can put numbers into two different categories. These categories are prime and composite. The number of factors that a number has directly affects whether or not the number is considered a prime number or a composite number.

What is a prime number?

Prime numbers are special numbers. As you can see in the text box, a prime number has only two factors. You can only multiply one and the number itself to get a prime number.

Example

Think about 13. Is it a prime number?

Yes. You can only get thirteen if you multiply 1 and 13. Therefore it is prime.

Here is a chart of prime numbers.

Notice that 1 has a next to it. One is neither prime nor composite.

What is a composite number?

A composite number is a number that has more than two factors. Most numbers are composite numbers. We can see from the chart that there are 25 prime numbers between 1 and 100. The rest are composite because they have more than two factors.

EXAMPLE 1

EXAMPLE 2

READ: Write the Prime Factorization Using a Factor Tree

Write the Prime Factorization of Given Numbers Using a Factor Tree

We can combine factoring and prime numbers together too. This is called prime factorization. When we factored numbers before, we broke down the numbers into two factors. These factors may have been prime numbers and they may have been composite numbers. It all depended on the number that we started with.

Example

Factor 36

36 can factor several different ways, but let’s say we factor it with 6 6.

These two factors are not prime factors. However, we can factor 6 and 6 again.

3 and 2 are both prime numbers.

When we factor a number all the way to its prime factors, it is called prime factorization.

It is a little tricky to keep track of all of those numbers, so we can use a factor tree to organize. Let’s organize the prime factorization of 36 into a factor tree.

Notice that we write 36 as a product of its primes.

Is there any easier way to write this?

Yes, we can use exponents for repeated factors. If you don’t have any repeated factors, you just leave your answer alone.

The prime factorization of 36 is .

EXAMPLE 1

EXAMPLE 2

CHECK Yourself! Prime Factorization