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, you'll see they are super useful and fun, like solving a cool puzzle.

We're going to learn how to find what numbers have in common. This is a great skill for everything from sharing sweets fairly with friends to figuring out when you'll next have football practice on the same day as your friend's music lesson. Let's get started!


Quick Review: What Are Multiples and Factors?

Before we find what's 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 another 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 "skip count".

Example: Let's find the multiples of 4.

4 x 1 = 4
4 x 2 = 8
4 x 3 = 12
4 x 4 = 16
...and so on!

So, the multiples of 4 are: 4, 8, 12, 16, 20, 24, ... (the list goes on forever!)

Now, What About Factors?

A factor is a number that divides another number perfectly, with no leftovers (no remainder). You can also think of factors as the numbers you multiply together to get a certain number.

Example: Let's find the factors of 12.

We think: "What numbers can I multiply to get 12?"

1 x 12 = 12
2 x 6 = 12
3 x 4 = 12

So, the factors of 12 are: 1, 2, 3, 4, 6, and 12. Factors are a complete list; they don't go on forever.

Key Takeaway

Multiples are found by Multiplying (they get bigger).
Factors are the numbers that can Fit perfectly 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 you and your best friend both know!

How to Find Common Factors (The Listing Method)

This is a super easy, step-by-step way to find all the common factors.

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 are: 1, 2, 3, 6, 9, 18

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

Step 3: Look at both lists and circle or write down the numbers that are in BOTH lists.
The numbers in both lists are: 1, 2, 3, and 6.

That's it! The common factors of 18 and 24 are 1, 2, 3, and 6.

The Biggest of Them All: The Highest Common Factor (H.C.F.)

The Highest Common Factor is exactly what it sounds like: it's the biggest or highest number from the list of common factors. We often write it as H.C.F. for short.

Example: Let's find the H.C.F. of 18 and 24.

We already found the common factors: 1, 2, 3, and 6.
Now, we just look for the biggest number in that list. The biggest number is 6!

So, the H.C.F. of 18 and 24 is 6.

Real-World Puzzle!

You have 18 chocolate bars and 24 lollipops. You want to make party bags for your friends that are all exactly the same, with no sweets left over. What is the biggest number of party bags you can make?
Answer: The H.C.F.! You can make 6 party bags. Each bag would have 3 chocolate bars (18 ÷ 6) and 4 lollipops (24 ÷ 6). Cool, right?

Key Takeaway

To find the H.C.F., first you list all the factors for both numbers, find the ones they have in common, and then pick the biggest one!


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 numbers.

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

How to Find Common Multiples (The Listing Method)

Example: Let's find some common multiples of 3 and 5.

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

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

Step 3: Find the numbers that are in BOTH lists.
The first two numbers we see in both lists are 15 and 30.

The common multiples of 3 and 5 are 15, 30, 45, ... and the list keeps going!

The First in Line: The Least Common Multiple (L.C.M.)

Since the list of common multiples goes on forever, we can't find the "highest" one. But we can find the smallest one! The Least Common Multiple is the smallest number that appears in both lists. We write it as L.C.M. for short.

Example: Let's find the L.C.M. of 3 and 5.

We found that the common multiples are 15, 30, 45, ...
What's the smallest, or "least," number in that list? It's 15!

So, the L.C.M. of 3 and 5 is 15. (This means our blinking lights will flash together for the first time after 15 seconds!)

Did you know?

The L.C.M. of two numbers will always be the same as or bigger than the largest of the two numbers. For example, the L.C.M. of 3 and 5 is 15, which is bigger than 5. This is a great way to check your answer!

Key Takeaway

To find the L.C.M., you list the multiples for both numbers, find the first one they have in common, and that's your answer!


A Super Shortcut: The Short Division Method

Listing all the factors and multiples is great, but what if the numbers are big? It could take a long time! Don't worry, there's a fast and clever method called short division.

Let's find the H.C.F. and L.C.M. of 24 and 36 using this method.

Finding the H.C.F. with Short Division

Step 1: Write the two numbers down. Draw an upside-down division symbol around them.

Step 2: Find a common factor that can divide BOTH numbers. Let's use 2.
2 | 24 36
Divide both numbers by 2 and write the answers below.
2 | 24 36
| 12 18

Step 3: Repeat! Can we divide 12 and 18 by a common factor? Yes, by 2 again!
2 | 24 36
2 | 12 18
| 6 9

Step 4: Repeat again! Can we divide 6 and 9 by a common factor? Not by 2, but we can use 3!
2 | 24 36
2 | 12 18
3 | 6 9
| 2 3

Step 5: Stop when there are no more common factors (except 1). We can't divide 2 and 3 by the same number.

To find the H.C.F., just multiply the numbers on the LEFT side.
H.C.F. = 2 x 2 x 3 = 12

Finding the L.C.M. with Short Division

Great news! We use the exact same steps. We just do one different thing at the end.

Let's use our finished short division from above:
2 | 24 36
2 | 12 18
3 | 6 9
| 2 3

To find the L.C.M., multiply the numbers on the LEFT side AND the numbers at the BOTTOM. It makes an 'L' shape!
L.C.M. = (2 x 2 x 3) x (2 x 3) = 72

Memory Trick!

H.C.F. = Multiply the numbers Hanging down the left.
L.C.M. = Multiply the numbers on the Left and the Lower bottom row.

Key Takeaway

Short division is a fast way to find both the H.C.F. and L.C.M. at the same time. Just remember which numbers to multiply for each one!