Greatest Common Factors and Least Common Multiples

What is a factor? What is a multiple?

Have you ever had a box of chocolates, and you tell yourself that you'll just have $2$ each day? That way, if there are 12 chocolates, you can make them last $6$ days ๐.

But, self-control is hard, so maybe instead you'll have $3$ a day and make them last $4$ days. Or, honestly, just have $12$ in $1$ day. Tomorrow's a new day โ๏ธ.

These are factor pairs! We can get to 12 in three different ways:

$121212โ=1ร12=2ร6=3ร4โ$

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

On the other hand, if we wanted to buy more boxes of chocolates, we could buy 1 box ($12$ chocolates), 2 boxes ($24$ chocolates), 3 boxes ($36$ chocolates), and so on.

So, the multiples of 12 are $12,24,36,...$

Notice, we can keep adding 12 to get more multiples, but we can't get more factors. To remember the difference, just think more multiples and few factors!

GCF & LCM Calculator

What is the

of

and

?

Use the

We're looking for the smallest multiple that these two numbers share. Think of it as a race! ๐ข ๐

Click the โStartโ button below to start generating multiples for each number. When you see a common multiple - a number that appears on both sides - click "Stop."

๐

๐

The LCM is

Not quite, look for the multiple that appears in both tables - hint: it's green!

That's right! Our LCM is ๐ฅ!

We can use the formula below โฌ๏ธ to calculate LCM.

$LCMย ofย (,)=GCFย ofย (,)รโ$

Step 1: Find the GCF

From our formula, we can see that we need the GCF, or greatest common factor, of our two numbers. So let's get that first.

To find the GCF, we need to find the factors of and .

We need the greatest common factor, so select the matching factor between the two numbers that is the BIGGEST.

๐ ๐

๐ ๐

Step 2: Plug-In the Formula

That's right! Our GCF is $1$. So, we have everything we need for our formula now - plug everything in below โฌ๏ธ

$LCM=$

$ร$

That's right! When you're ready, click the ๐ below to get the final, calculated LCM.

Step 3: Calculate

When we do the math, we find that our LCM is:

$LCM(,)โ=1รโ=0โ$

Let's start by getting all the factors of each of our numbers. We'll use the U-Turn method to find the pairs of factors.

๐ ๐

๐ ๐

Now, pick the BIGGEST factor that both numbers have in common.

That's right! Our greatest common factor, or GCF, is .

Recap ๐งข GCF & LCM

Remember, GCF stands for Greatest Common Factor. Follow these steps to find the GCF of two numbers:

Step 1: Get All Factor Pairs

First, find all the factors for both your numbers.

Step 2: Find the Greatest Common Factor

Now, find the biggest factor that both numbers have in common - that's your GCF!

Remember, LCM stands for Least Common Multiple. One way to find the LCM of two numbers is to just list out multiples:

Step 1: List out Multiples

List out the multiples of each number, alternating one for each number.

Step 2: Stop ๐

Stop as soon as you see a multiple in common between your two numbers - that's the LCM!

Remember, LCM stands for Least Common Multiple. One way to find the LCM of two numbers is to use a formula:

Step 1: Get the GCF

First, you'll need the GCF of the two numbers. List out the factors for both numbers, and find the biggest factor they have in common.

Step 2: Plug into the Formula ๐

Plug everything into this formula:

$LCM=GCF1stย numberร2ndย numberโ$

Calculate it out to get your LCM!

No sweat if this doesn't all make sense yet. Click below for more help!

What is a factor?

Letโs say we decide to host a Mardi Gras partyโon a budget. ๐๐๐

We can use factors to plan out how many people we can invite, based on the amount of food we have:

24 slices of king cake ๐

18 servings of jambalaya ๐ฝ๏ธ

If we want everyone to get an equal number of king cake slices and an equal amount of jambalaya, whatโs the maximum number of guests we can invite?

To answer this question, we need to use factors.

Factors are the pairs of numbers that multiply to form a given number: for example, the factors of 24 are 1 & 24, 2 & 12, 3 & 8, and 6 & 4.

What are the factors of 18? Select the matching factor pairs:

We need the number of guests to be a factor of both 24 and 18. And, if we want the most guests we can have, we want the largest factor they share.

Hereโs a Venn diagram of the factors for 18 and 24. Click on the largest one they have in common:

18

24

That's right! โจ If we invite 6 people, each person gets 4 slices of king cake and 3 serving of jambalaya.

6 is the Greatest Common Factor (GCF)โAKA, the largest factor two numbers have in common - of 18 and 24.

But, using a Venn diagram isnโt the fastest way to find the GCF. Keep scrolling to try out our favorite method: the U-Turn Method!

How do you calculate the GCF?

The U-Turn Method is a trick we can use to get all the factors. So, channel your inner Olivia Rodrigoโletโs get our GCF driverโs license!

Letโs find the greatest common factor of 12 and 30.

Weโll fill in each row of the tables below with a factor pair: starting with 1 on the left side, which goes into $12ย12$ times, and $30ย30$ times.

๐ 12 ๐

1

12

2

6

3

4

4

3

โฉ๏ธ

๐ 30 ๐

1

30

2

15

3

10

5

6

6

5

โฉ๏ธ

Nice work! So to find our greatest common factor, or GCF, we need to find all our factors, then identify the common factors, and then find the largest one.

A Guide to Multiples

Multiples are what we get after we multiply a number by an integer (not a fraction): for example, the multiples of 3 are 3, 6, 9, 12, 15, and so on!

Try entering in the first five multiples of 6:

You can use multiples to find the Least Common Multiple (LCM), which is the smallest multiple two numbers have in common.

Let's say we're making some Mardi Gras necklaces with different color beads for our party.

At the store, green beads are sold in packs of 8, and purple beads in packs of 12. We can find the least common multiple of 8 and 12 to figure out the fewest number of packs we need to buy to have an equal number of green and purple beads.

How do you calculate the LCM?

There are two main methods for calculating LCM: the Finding Multiples method and the GCF Formula method. Letโs start by learning the Finding Multiples method!

You can think of the Finding Multiples method as a race. No matter how many multiples you have to go through, sooner or later both numbers will cross the finish line and land on the LCM!

Letโs practice listing multiples now by filling in these tables, row by row, until we find a match in order to find the least common multiple of 8 and 12!

๐ 8 ๐

8

๐ 12 ๐

12

We start with the number itself in each table, then keep adding the number to the previous row.

๐ 8 ๐

8

16

24

๐ 12 ๐

12

24

Between the two tables, the first match (AKA smallest match) we have is $24$โso thatโs our LCM! We made it to the finish line! ๐ข ๐

Thatโs not the only method you can use to find the LCM, though. You can actually use the GCF of two numbers to calculate the LCM of those numbers (letโs call them a and b) using this formula:

$LCM(a,b)=GCF(a,b)aรbโ$

This formula looks complicated, but itโs actually not so bad! To find the LCM, all you have to do is multiply the two numbers together, and divide them by their GCF.

Letโs try it out with our same example numbers: 8 and 12. The greatest common factor of 8 and 12, which we can find using the U-Turn method, is 4: