Factors

Are Prime Numbers Useful in Real Life?

Some students ask this question in class.

“Teacher… why are we learning prime numbers?”

You might be thinking:

“When will I ever use this?”

That is a good question.

Let us think about something you do every day.

You open your phone and:

• send a WhatsApp message
• log in to Instagram
• watch YouTube
• your parent sends money using mobile banking

All these things happen every day.

But here is something interesting.

When you send a message or money, the information must be protected so that bad people cannot see it or steal it.

So the computer hides the information.

It turns the message into a secret code.

And mathematics helps the computer create that secret code.

One of the things used in that mathematics is:

Prime numbers.

Yes.

The same prime numbers you are learning in class.

Think of It Like a Padlock 🔒

Imagine you have a padlock on your bag.

The lock protects the things inside your bag.

Only someone with the correct key can open it.

In the digital world, computers also need locks to protect information.

But instead of using a metal lock, computers use mathematics.

Prime numbers help create those digital locks.

Why Prime Numbers?

Prime numbers are special numbers.

Example:

2

3

5

7

11

They can only be divided by 1 and themselves.

Because of this special property, they help computers create very strong digital locks.

Breaking these locks is extremely difficult.

Something Amazing

The prime numbers you see in school are small.

But computers use very, very big prime numbers.

Some are so big they can have hundreds of digits.

These big prime numbers help protect:

• bank money
• passwords
• online messages

So every time you use the internet, mathematics is quietly helping keep things safe.

The Important Lesson

So when you learn prime numbers, you are not learning something useless.

You are learning part of the mathematics that helps protect:

• phones
• computers
• banks
• the internet

Mathematics is one of the reasons modern technology works safely.

Now, let’s break down the topic and understand it step by step.

TOPIC: FACTORS

Subtopic: Factors of Composite Numbers and Prime Factors

Today we are going to learn something called Factors.

But before we talk about factors, we must first understand something very small.

That small idea is called sharing equally.

Step 1: Understanding Sharing Equally

Imagine you have 4 sweets.

And you have 2 children.

You want to give each child the same number of sweets.

So you share them.

Child 1 gets 2 sweets

Child 2 gets 2 sweets

Now all the sweets are finished.

Nothing is left.

When we share things like this, we say we divide.

So we say:

4 ÷ 2 = 2

This means:

4 sweets shared between 2 children gives 2 sweets each.

Step 2: What If Something Remains?

Now imagine you have 5 sweets and 2 children.

You try to share them equally.

Child 1 gets 2

Child 2 gets 2

But 1 sweet remains.

That means the sharing was not exact.

We say:

5 cannot be divided by 2 exactly.

Because something remained.

Step 3: Understanding Exact Division

When we say divide exactly, we mean:

sharing equally

nothing remains

Example:

6 sweets shared between 3 children.

Each child gets 2.

Nothing remains.

So:

6 ÷ 3 = 2

This is exact division.

Step 4: Now We Are Ready to Learn Factors

A factor is a number that can divide another number exactly.

That means:

when you divide, nothing remains.

Let Us Find the Factors of 6

We ask:

Which numbers can divide 6 exactly?

We test numbers slowly.

Can 1 divide 6?

6 ÷ 1 = 6

Nothing remains.

So 1 is a factor of 6.

Can 2 divide 6?

6 ÷ 2 = 3

Nothing remains.

So 2 is a factor of 6.

Can 3 divide 6?

6 ÷ 3 = 2

Nothing remains.

So 3 is a factor of 6.

Can 4 divide 6?

6 ÷ 4

4 goes into 6 one time.

But 2 remains.

So 4 is not a factor.

Can 5 divide 6?

6 ÷ 5

1 remains.

So 5 is not a factor.

Can 6 divide 6?

6 ÷ 6 = 1

Nothing remains.

So 6 is a factor.

Final Answer

The factors of 6 are:

1

2

3

6

Step 5: Understanding This With Real Life

Imagine a teacher has 6 learners.

The teacher wants to arrange them into equal rows.

Possible rows are:

1 row of 6 learners

2 rows of 3 learners

3 rows of 2 learners

6 rows of 1 learner

The numbers used are:

1, 2, 3, 6

These are the factors of 6.

Step 6: Every Number Has Two Special Factors

Every number always has these two factors:

1

and

the number itself

Example:

Factors of 5

1 and 5

Factors of 10

1 and 10

Why?

Because:

every number can be divided by 1

every number can be divided by itself

Step 7: Prime Numbers

Now let us learn something new.

Some numbers have only two factors.

These factors are:

1

and

the number itself.

These numbers are called Prime Numbers.

Example: Number 5

Let us test.

5 ÷ 1 = 5

5 ÷ 2 = not exact

5 ÷ 3 = not exact

5 ÷ 4 = not exact

5 ÷ 5 = 1

Only two factors:

1 and 5

So 5 is a prime number.

Other Prime Numbers

2

3

5

7

11

13

These numbers cannot be divided by any other number except 1 and themselves.

Step 8: Composite Numbers

Some numbers have many factors.

More than two factors.

These numbers are called Composite Numbers.

Example: Number 8

Let us test.

8 ÷ 1 = 8

8 ÷ 2 = 4

8 ÷ 4 = 2

8 ÷ 8 = 1

Factors are:

1, 2, 4, 8

That is more than two factors.

So 8 is a composite number.

Another Example: Number 12

Factors of 12:

1

2

3

4

6

12

That is many factors.

So 12 is composite.

Step 9: Difference Between Prime and Composite

Example:

7 → Prime

10 → Composite

Step 10: Finding Prime Factors

First let us remember something very important.

A number can sometimes be written as two numbers multiplied together.

Example:

12 can be made by multiplying numbers.

Let us try some multiplications we already know.

1 × 12 = 12

2 × 6 = 12

3 × 4 = 12

So we now know:

12 can be written as

12 = 1 × 12

12 = 2 × 6

12 = 3 × 4

These are called factor pairs.

Which pair should we use?

When we want prime factors, we look for a pair where one number is a prime number.

Look at the pairs again.

1 × 12

2 × 6

3 × 4

Let us check them.

1 is not prime.

12 is not prime.

So this pair is not helpful.

Now look at:

2 × 6

2 is prime.

So this is a good place to start.

So we write:

12 = 2 × 6

But we are not finished

We want prime numbers only.

So we must check 6.

Is 6 a prime number?

Let us test.

6 ÷ 1 = 6

6 ÷ 2 = 3

6 ÷ 3 = 2

6 ÷ 6 = 1

6 has four factors.

So 6 is not prime.

That means we must break 6 again.

Breaking 6 into smaller numbers

We now look for numbers that multiply to give 6.

1 × 6 = 6

2 × 3 = 6

Now check:

2 is prime

3 is prime

So we stop.

Now we combine everything.

12 = 2 × 6

6 = 2 × 3

So

12 = 2 × 2 × 3

These are the prime factors of 12.

Step 11: Another Example (18)

Let us repeat the same thinking.

First ask:

What numbers multiply to give 18?

Try multiplication you already know.

1 × 18 = 18

2 × 9 = 18

3 × 6 = 18

So 18 can be written as:

18 = 1 × 18

18 = 2 × 9

18 = 3 × 6

Now we choose a pair where one number is prime.

Look at:

2 × 9

2 is prime.

So we write:

18 = 2 × 9

Now we check 9.

Is 9 prime?

Test:

9 ÷ 1 = 9

9 ÷ 3 = 3

9 ÷ 9 = 1

9 has three factors.

So it is not prime.

So we break 9 again.

What multiplies to give 9?

1 × 9

3 × 3

3 is prime.

So

9 = 3 × 3

Now combine everything.

18 = 2 × 9

9 = 3 × 3

So

18 = 2 × 3 × 3

Prime factors are:

2, 3, 3

Step 12: One More Example (20)

First ask:

What numbers multiply to give 20?

1 × 20

2 × 10

4 × 5

Let us choose:

2 × 10

Because 2 is prime.

So:

20 = 2 × 10

Now check 10.

Is 10 prime?

Test:

10 ÷ 1 = 10

10 ÷ 2 = 5

10 ÷ 5 = 2

10 ÷ 10 = 1

So 10 has more than two factors.

It is not prime.

So we break 10.

Numbers that multiply to give 10:

1 × 10

2 × 5

Both are prime.

So:

10 = 2 × 5

Now combine everything.

20 = 2 × 10

10 = 2 × 5

So

20 = 2 × 2 × 5

Step 13: Using a Factor Tree (Visual Method)

A factor tree is just a drawing that helps us break numbers slowly.

Let us do 24.

First ask:

What numbers multiply to give 24?

1 × 24

2 × 12

3 × 8

4 × 6

We can choose 2 × 12.

Because 2 is prime.

So we draw:

24 ↙ ↘ 2 12

Now break 12.

We already know:

12 = 2 × 6

So draw:

12 ↙ ↘ 2 6

Now break 6.

6 = 2 × 3

So draw:

6 ↙ ↘ 2 3

Now look at the bottom numbers.

2

2

2

3

All of them are prime numbers.

So we stop.

Therefore:

24 = 2 × 2 × 2 × 3

These are the prime factors of 24.

Important Thinking Rule (for Students)

Whenever you find prime factors:

Start with the number.

Ask: what numbers multiply to give this number?

Choose a pair where one number is prime.

Keep breaking numbers until all numbers are prime.

Then stop.

Step 14: Why This Topic Is Important

Factors help us later in mathematics when we learn:

HCF (Highest Common Factor)

LCM (Lowest Common Multiple)

Simplifying fractions

Algebra

Without understanding factors, these topics become difficult.

Step 15: Practice Questions

Easy

Write the factors of 6

Write the factors of 8

Write the factors of 10

Medium

State whether 7 is prime or composite

State whether 9 is prime or composite

Find the prime factors of 12

Harder

Find the prime factors of 18

Find the prime factors of 20

Find the prime factors of 24

Final Teacher Summary

Today we learned:

A factor is a number that divides another number exactly.

A prime number has only two factors.

A composite number has more than two factors.

Prime factors are prime numbers that multiply together to form another number.

Example:

12 = 2 × 2 × 3

If you didn’t understand the prime tree, you can use the method below.

The Division Ladder Method (Prime Factor Ladder)

Think of it like climbing down a ladder.

Each step you divide the number by a prime number.

Example 1: Prime Factors of 12

We start with 12.

We ask:

What is the smallest prime number that can divide 12 exactly?

The smallest prime number is 2.

Check:

12 ÷ 2 = 6

So we write:

2 | 12
    6

Now we repeat the question.

What is the smallest prime number that can divide 6?

Again it is 2.

6 ÷ 2 = 3

So we continue the ladder:

2 | 12
2 |  6
     3

Now we check 3.

Is 3 a prime number?

Yes.

So we divide by 3.

3 ÷ 3 = 1

Now the ladder looks like this:

2 | 12
2 |  6
3 |  3
     1

Now we stop because we reached 1.

The numbers on the left side are the prime factors.

So:

12 = 2 × 2 × 3

Example 2: Prime Factors of 18

Start with 18.

Smallest prime number is 2.

18 ÷ 2 = 9

2 | 18
    9

Now ask:

Can 2 divide 9?

No.

So we try the next prime number.

3

9 ÷ 3 = 3

2 | 18
3 |  9
     3

Now divide again.

3 ÷ 3 = 1

2 | 18
3 |  9
3 |  3
     1

Prime factors are the numbers on the left.

18 = 2 × 3 × 3

Example 3: Prime Factors of 20

Start with 20.

Smallest prime number is 2.

20 ÷ 2 = 10

2 | 20
    10

Again divide by 2.

10 ÷ 2 = 5

2 | 20
2 | 10
     5

Now divide by 5.

5 ÷ 5 = 1

2 | 20
2 | 10
5 |  5
     1

Prime factors:

20 = 2 × 2 × 5

Example 4: Prime Factors of 24

Start with 24.

24 ÷ 2 = 12

2 | 24
    12

12 ÷ 2 = 6

2 | 24
2 | 12
     6

6 ÷ 2 = 3

2 | 24
2 | 12
2 |  6
     3

3 ÷ 3 = 1

2 | 24
2 | 12
2 |  6
3 |  3
     1

Prime factors:

24 = 2 × 2 × 2 × 3