Hello, Super Mathematicians!

Welcome to the amazing world of Common Multiples and Common Factors! Don't worry if those words sound a bit big. By the end of this chapter, you'll see they are super useful and fun, like solving a cool puzzle.

We are going to learn how to find what numbers have in common. This is a great skill for everything from sharing items fairly with friends to figuring out patterns in numbers. Let's get started!


Quick Review: What Are Multiples and Factors?

Before we find what is common, let's remember what multiples and factors are. Think of them as a family – they are related but different!

Let's Talk About Multiples!

A multiple of a number is what you get when you multiply it by a whole number. A simple way to think about it is that multiples are the answers in a number's times table, or what you get when you count by that number.

Example: Let's list the first 6 multiples of 4.

\(4 \times 1 = 4\)
\(4 \times 2 = 8\)
\(4 \times 3 = 12\)
\(4 \times 4 = 16\)
\(4 \times 5 = 20\)
\(4 \times 6 = 24\)

So, the first 6 multiples of 4 are: 4, 8, 12, 16, 20, and 24.

How to check if a number is a multiple:
Is 35 a multiple of 5? Yes, because \(35 \div 5 = 7\) with no remainder!

Now, What About Factors?

A factor of a whole number divides that number exactly, with no remainder. You can also think of factors as the numbers you multiply together to get a certain product.

Example: Let's find all the factors of 12.

We think of multiplication pairs that equal 12:

\(1 \times 12 = 12\)
\(2 \times 6 = 12\)
\(3 \times 4 = 12\)

So, the factors of 12 are: 1, 2, 3, 4, 6, and 12.

How to check if a 1-digit number is a factor:
Is 3 a factor of 48? Divide: \(48 \div 3 = 16\). Since there is no remainder, 3 is a factor of 48!

Key Takeaway

Multiples are found by Multiplying (they get bigger).
Factors are the numbers that Fit exactly into another number (they are smaller or equal).


Finding What's in Common: Common Factors

When two numbers share the same factor, we call it a common factor. "Common" just means "shared". It's like having a common friend – a friend that both you and your classmate know!

How to Find Common Factors (The Listing Method)

Here is a step-by-step way to find all the common factors of two numbers up to 100.

Example: Let's find the common factors of 18 and 24.

Step 1: List all the factors of the first number (18).
Factors of 18: 1, 2, 3, 6, 9, 18

Step 2: List all the factors of the second number (24).
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Step 3: Look at both lists and find the numbers that appear in BOTH lists.
The numbers in both lists are: 1, 2, 3, and 6.

The common factors of 18 and 24 are 1, 2, 3, and 6.

Note: The number 1 is always a common factor of any two whole numbers!


Let's Count Together: Common Multiples

Just like with factors, numbers can also share multiples. A common multiple is a number that is a multiple of two or more 1-digit numbers.

Imagine two blinking lights. One blinks every 3 seconds, and the other blinks every 5 seconds. A common multiple tells you when they will blink at the exact same time!

How to Find Common Multiples (The Listing Method)

Example: Let's find the first two common multiples of 3 and 5.

Step 1: List the multiples of the first number (3).
Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, ...

Step 2: List the multiples of the second number (5).
Multiples of 5: 5, 10, 15, 20, 25, 30, 35, 40, ...

Step 3: Find the numbers that appear in BOTH lists.
The first two shared numbers are 15 and 30.

The first two common multiples of 3 and 5 are 15 and 30.

Key Takeaway

To find common multiples, list the multiples of each 1-digit number in order, then spot the numbers that are shared in both lists!