Hello Math Detectives!

Welcome to a super exciting adventure into the world of numbers! Today, we're going to uncover the secret building blocks that make up every number you know. We'll learn about Factors, Multiples, Prime Numbers, and Composite Numbers.

Understanding these ideas is like getting a secret key to unlock lots of math puzzles. It will help you with multiplication, division, and much more. Let's get started!


Section 1: Factors - The Building Blocks of Numbers

What are Factors?

Imagine you have 12 cookies and you want to arrange them in equal rows. How could you do it?

- You could have 1 row of 12 cookies. (1 x 12)
- You could have 2 rows of 6 cookies. (2 x 6)
- You could have 3 rows of 4 cookies. (3 x 4)

The numbers 1, 2, 3, 4, 6, and 12 are the Factors of 12! A factor is a whole number that divides another number exactly, with no remainder left over.

Think of them like Lego blocks. The factors are the smaller blocks you use to build a bigger number!

How to Find All the Factors of a Number

Don't worry if this seems tricky at first! Here is an easy step-by-step way to find all the factors of a number. Let's try finding the factors of 18.

Step 1: Start with 1.

Always begin with 1. One times the number itself is always the first pair of factors.
Example: $$1 \times 18 = 18$$. So, 1 and 18 are factors.

Step 2: Try 2.

Ask yourself: "Can 18 be divided by 2 with no remainder?" Yes!
Example: $$2 \times 9 = 18$$. So, 2 and 9 are factors.

Step 3: Try 3.

Ask yourself: "Can 18 be divided by 3 with no remainder?" Yes!
Example: $$3 \times 6 = 18$$. So, 3 and 6 are factors.

Step 4: Keep going... Try 4.

Can 18 be divided by 4? No, there will be a remainder. So, 4 is not a factor.

Step 5: Try 5.

Can 18 be divided by 5? No, there will be a remainder. So, 5 is not a factor.

Step 6: Stop when the numbers meet or repeat.

The next number to try is 6, but we already have it in our list from Step 3! This is our signal to stop.

Step 7: List all the factors.

Now, just list all the numbers we found, from smallest to biggest.
The factors of 18 are: 1, 2, 3, 6, 9, 18.

Key Takeaway for Factors

Factors are numbers that can be multiplied together to get a certain number. Every whole number greater than 1 has at least two factors: 1 and itself.


Section 2: Multiples - Skip-Counting Champions!

What are Multiples?

A multiple is the result of multiplying a number by a whole number (like 1, 2, 3, 4, and so on). The easiest way to think about it is skip-counting or your times tables!

Example: Let's find the multiples of 4.
$$4 \times 1 = 4$$
$$4 \times 2 = 8$$
$$4 \times 3 = 12$$
$$4 \times 4 = 16$$

So, the first few multiples of 4 are 4, 8, 12, 16, and so on.

Did you know?

A number has an endless (infinite) list of multiples! You could keep finding multiples forever and ever.

Key Takeaway for Multiples

Multiples are the numbers you get by skip-counting or by multiplying a number by 1, 2, 3, etc. They are the answers in a times table!


Section 3: Factors and Multiples - A Perfect Pair!

How Are They Connected?

Factors and multiples are opposites, like heads and tails on a coin. They work together.

Let's look at the equation: $$3 \times 5 = 15$$

From this one equation, we can say two things:
1. 3 and 5 are factors of 15.
2. 15 is a multiple of 3 and a multiple of 5.

See? They are connected! If a number is a multiple of another, then that other number must be its factor.

Key Takeaway for the Connection

Factors are the builders, and the multiple is what they build. They are two parts of the same multiplication story.


Section 4: Special Kinds of Numbers

Now that we know about factors, we can sort numbers into two very special groups: Prime and Composite.

Prime Numbers

A prime number is a whole number greater than 1 that has exactly two factors: 1 and itself.

They can't be divided evenly by any other numbers. They are unique!

Examples of prime numbers:
2 (factors are 1, 2)
3 (factors are 1, 3)
5 (factors are 1, 5)
7 (factors are 1, 7)
11 (factors are 1, 11)

Did you know?

The number 2 is the ONLY even prime number. Every other even number can be divided by 2, so it will have more than two factors.

Composite Numbers

A composite number is a whole number greater than 1 that has more than two factors.

They are "composed" of other factors.

Examples of composite numbers:
4 (factors are 1, 2, 4)
6 (factors are 1, 2, 3, 6)
9 (factors are 1, 3, 9)
10 (factors are 1, 2, 5, 10)
12 (factors are 1, 2, 3, 4, 6, 12)

What about the number 1?

This is a great question! The number 1 is very special. It is NEITHER prime NOR composite. Why? Because it only has ONE factor: itself! To be prime, it needs exactly two factors.

Quick Review Box

- A number with exactly 2 factors is PRIME.
- A number with more than 2 factors is COMPOSITE.
- The number 1 is NEITHER.

Key Takeaway for Prime and Composite

We can sort numbers based on how many factors they have. This helps us understand what makes each number special.


Section 5: How to Find Prime Numbers - The Sieve of Eratosthenes

An ancient Greek mathematician named Eratosthenes came up with a brilliant way to find all the prime numbers up to 100. It's like using a sieve in the kitchen to separate what you want to keep! It's fun and easy. Let's try it!

Step-by-Step Guide to Using the Sieve

Imagine you have a chart with all the numbers from 1 to 100.

Step 1: Get Rid of 1

Cross out the number 1. We know it's not prime.

Step 2: Circle 2, Cross Out its Multiples

Circle the number 2 – it's our first prime number! Now, go through the chart and cross out all the other multiples of 2 (4, 6, 8, 10, ... all the way to 100). This removes all the even numbers.

Step 3: Circle 3, Cross Out its Multiples

The next number on the chart that isn't crossed out is 3. Circle it! Now, cross out all the multiples of 3 (6, 9, 12, 15...). Some will already be crossed out, and that's okay!

Step 4: Circle 5, Cross Out its Multiples

Find the next available number. It's 5! Circle the 5 and cross out all of its multiples (10, 15, 20, 25...).

Step 5: Circle 7, Cross Out its Multiples

The next number that isn't crossed out is 7. Circle it, and cross out all its multiples (14, 21, 28, 35...).

The Final Step!

Continue this process until every number on the chart is either circled or crossed out. All the circled numbers are the prime numbers up to 100! You did it!

Key Takeaway for the Sieve

The Sieve of Eratosthenes is a clever and visual way to find all prime numbers by systematically removing the composite numbers.