How to Find the LCM (2 Different Ways) | Least Common Multiple | Math with Mr. J

Summary

In this video, Mr. J explains how to find the least common multiple (LCM) using two methods: listing multiples and prime factorization. He begins with listing multiples, which involves skip counting until a common multiple is found. This method works well for smaller numbers, as shown in examples with 9 and 12, and 10 and 25. For larger numbers, Mr. J demonstrates using prime factorization, breaking numbers down into their prime factors and aligning them to find the LCM. He provides clear, step-by-step explanations and examples to help viewers understand both strategies.

Takeaways

• 🔢 The least common multiple (LCM) is the smallest multiple that two or more numbers share.
• 🧮 A multiple is the result of multiplying a given number by an integer.
• 🔄 One method to find the LCM is by listing out multiples of each number and finding the smallest common multiple.
• 9️⃣ To find the LCM of 9 and 12, the multiples of 9 are 9, 18, 27, 36, and 45, while the multiples of 12 are 12, 24, 36, 48, and 60. The LCM is 36.
• 5️⃣ The LCM of 10 and 25 is found by listing their multiples, 50 being the smallest common multiple.
• 🔍 Another method to find the LCM is using prime factorization, which is more efficient with larger numbers.
• 🧩 In prime factorization, break down each number into its prime factors and match them vertically.
• 📊 For example, the prime factors of 15 are 3 and 5, and the prime factors of 27 are three 3's, making the LCM 135.
• 🔗 For 28 and 52, the prime factors are 2x2x7 for 28 and 2x2x13 for 52. The LCM is 364.
• ✔️ Both strategies (listing multiples and prime factorization) can be useful for finding the LCM depending on the complexity of the numbers involved.

Q & A

What are two strategies mentioned in the video for finding the Least Common Multiple (LCM)?

The two strategies mentioned are listing out multiples and using prime factorization.

How is a multiple defined in the context of finding the LCM?

A multiple is the result of multiplying a given number by an integer. Another way to think of it is by skip counting.

In the first example with the numbers 9 and 12, what is the least common multiple (LCM)?

The least common multiple (LCM) of 9 and 12 is 36.

How many multiples does the video suggest listing to find the LCM?

The video suggests listing four or five multiples of each number to find the LCM.

What is the least common multiple (LCM) of 10 and 25 according to the video?

The least common multiple (LCM) of 10 and 25 is 50.

Why might prime factorization be a better strategy for finding the LCM of larger numbers?

Prime factorization is helpful for larger numbers because listing multiples can be difficult and time-consuming as the numbers increase in value.

What is the prime factorization of 15 as explained in the video?

The prime factorization of 15 is 3 × 5.

What is the prime factorization of 27 as explained in the video?

The prime factorization of 27 is 3 × 3 × 3.

In the example with 15 and 27, what is the least common multiple (LCM) and how is it calculated?

The least common multiple (LCM) of 15 and 27 is 135. It is calculated by multiplying 3 × 5 × 3 × 3.

How does the video calculate the LCM of 28 and 52 using prime factorization?

The prime factorization of 28 is 2 × 2 × 7, and for 52 it is 2 × 2 × 13. The LCM is calculated as 2 × 2 × 7 × 13, which equals 364.

Outlines

00:00

📚 Introduction to Finding the Least Common Multiple (LCM)

In this section, Mr. J introduces the topic of finding the Least Common Multiple (LCM) using two strategies: listing multiples and prime factorization. He explains that the LCM is the smallest multiple that two numbers share, and clarifies the concept of multiples as the result of multiplying a number by integers. Mr. J also suggests that a simpler way to think about multiples is through 'skip counting.' He provides an example with numbers 9 and 12, listing their multiples and identifying the LCM as 36. He concludes by reminding viewers that multiples (and common multiples) are infinite.

05:03

🔢 Finding the LCM by Listing Multiples: Example with 10 and 25

Mr. J moves on to a second example to demonstrate how to find the LCM by listing multiples. This time, he uses the numbers 10 and 25. After listing the first five multiples of each number, he identifies 50 as the common multiple and the LCM. Mr. J then summarizes that listing multiples is a helpful strategy, but it can be time-consuming when working with larger numbers. He transitions to the next method, prime factorization, as a more efficient strategy for larger numbers.

10:06

🧮 Prime Factorization: Example with 15 and 27

Mr. J introduces the prime factorization method for finding the LCM, particularly useful for larger numbers. He walks through the prime factorization of 15 (3 × 5) and 27 (3 × 3 × 3). After listing the prime factors of both numbers vertically, he explains how to combine them by bringing down one representative from each column. In this example, multiplying the factors (3 × 5 × 3 × 3) results in an LCM of 135 for the numbers 15 and 27.

🔍 Prime Factorization: Example with 28 and 52

In this final example, Mr. J applies the prime factorization method to the numbers 28 and 52. He breaks 28 into 2 × 2 × 7 and 52 into 2 × 2 × 13, listing their prime factors vertically. He then multiplies the factors (2 × 2 × 7 × 13), resulting in an LCM of 364. Mr. J concludes the lesson by summarizing the two strategies (listing multiples and prime factorization) for finding the least common multiple, ensuring that viewers understand both methods.

Keywords

💡Least Common Multiple (LCM)

The least common multiple (LCM) is the smallest number that is a multiple of two or more numbers. It is central to the video's theme as it teaches viewers how to calculate the LCM of numbers using different methods. For example, in the video, the LCM of 9 and 12 is found to be 36 by listing multiples, and the LCM of 15 and 27 is found to be 135 using prime factorization.

💡Multiple

A multiple is the result of multiplying a given number by an integer. In the video, multiples are used to explain how to find common numbers that two or more numbers share. For instance, multiples of 9 are 9, 18, 27, 36, and multiples of 12 are 12, 24, 36, 48. The number 36 is a common multiple of both.

💡Prime Factorization

Prime factorization is the process of breaking down a number into its prime number factors. In the video, it is used as a strategy to find the LCM of larger numbers. For example, the prime factorization of 15 is 3 × 5, and for 27, it is 3 × 3 × 3. The LCM is then calculated using these prime factors.

💡Common Multiple

A common multiple is a number that is a multiple of two or more numbers. The video emphasizes finding the 'least' common multiple to solve problems. For instance, 36 is the smallest number that both 9 and 12 share as a multiple, making it the least common multiple.

💡Skip Counting

Skip counting is a method of counting forward by a number other than one, such as counting by 9s or 12s to find multiples. The video uses this to explain how multiples are derived and how listing multiples can lead to finding the LCM.

💡Factor

A factor is a number that divides evenly into another number. In prime factorization, the numbers are broken down into their prime factors to find the LCM. For example, 3 and 5 are factors of 15, and 3 is a factor of 27.

💡Prime Number

A prime number is a number greater than 1 that has no positive divisors other than 1 and itself. In the video, prime numbers like 3, 5, and 7 are used during the prime factorization method to find the LCM of different numbers.

💡Integer

An integer is a whole number, positive or negative, including zero. In the video, integers are used when explaining how multiples are formed, such as multiplying an integer by a number to generate its multiples.

💡Listing Multiples

Listing multiples is one of the methods shown in the video to find the LCM. It involves writing out the first few multiples of each number and identifying the smallest common one. For example, listing multiples of 9 (9, 18, 27, 36) and 12 (12, 24, 36) helps identify 36 as the LCM.

💡Endless Multiples

Endless multiples refer to the fact that multiples of a number go on infinitely. The video emphasizes that while only a few multiples are listed to find the LCM, there are infinitely many more. For example, after listing 9, 18, 27, 36, 45 for the number 9, it's noted that the sequence could continue indefinitely.

Highlights

Introduction to finding the least common multiple (LCM) using two strategies: listing multiples and prime factorization.

Explanation of multiples as numbers that result from multiplying a given number by an integer.

Simplified explanation of multiples through 'skip counting,' making it easier to understand and visualize.

Worked example: Finding the LCM of 9 and 12 using the method of listing multiples.

Strategy tip: When listing multiples, stop after 4 or 5 multiples to start comparing common multiples, but extend the list if necessary.

Detailed process for finding common multiples of 9 and 12, identifying 36 as the least common multiple.

Clarification that multiples (and common multiples) are infinite, meaning that the least common multiple is the smallest shared value, but there are more common multiples.

Worked example: Finding the LCM of 10 and 25 using the listing multiples strategy, with 50 as the LCM.

Introduction to prime factorization as a second strategy for finding the LCM, particularly useful for larger numbers.

Worked example: Using prime factorization to find the LCM of 15 and 27, resulting in 135 as the LCM.

Explanation of matching prime factors vertically to simplify finding the LCM.

Step-by-step multiplication process of the prime factors to calculate the LCM.

Worked example: Using prime factorization to find the LCM of 28 and 52, resulting in 364 as the LCM.

Importance of understanding both listing multiples and prime factorization as complementary strategies for finding the LCM.

Concluding remarks on the efficiency of prime factorization for larger numbers and summary of both methods covered.

Transcripts

welcome to math with Mr J

[Music]

in this video I'm going to cover how to

find the least common multiple the LCM

using two different strategies listing

out multiples and prime factorization we

will start with the strategy of listing

out some multiples of each number in

order to find the least common multiple

and then we'll move on to using prime

factorization now as far as the least

common multiple between numbers this is

going to be the smallest multiple in

value that both numbers share now a

multiple is the result of multiplying a

given number by an integer when we think

of the multiples of a number we need to

think about the numbers we get when

multiplying that given number by

integers a simpler way to think about

multiples is to think about skip

counting so all of the numbers something

is going to hit when you count count up

by that number those are all going to be

multiples this will make a lot more

sense as we go through our examples

let's jump into our examples starting

with number one where we have 9 and 12.

we're going to start by listing some

multiples of both 9 and 12. then we will

look for common multiples and

specifically the least common multiple

also referred to as the LCM let's start

with some multiples of nine which are

nine times one which is nine nine times

two is eighteen nine times three is

twenty-seven nine times four is

thirty-six and nine times five

is 45 so you can see that we just skip

counted by nine to list those multiples

9 18 27 36 45 so on and so forth now I

stopped at 45 because multiples go on

forever they are endless they are

infinite my suggestion is to list four

or five multiples when looking for the

least common multiple so list four or

five multiples for each number look for

any in common and if you don't have any

in common you can always extend the

multiples lists

now let's list the first five multiples

of twelve twelve times one is twelve

twelve times two is twenty four twelve

times three is thirty-six twelve times

four

is forty eight and twelve times five

is sixty so again you can see that we

skip counted there we skip counted by

twelve so 12 24 36 48 60 so on and so

forth now that we have some multiples

listed for both 9 and 12 we need to look

for any common multiples so any

multiples that they share and then

specifically we need to look for the

least common multiple well 36 is a

common multiple

and it's going to be the least common

multiple so the smallest multiple in

value that they share so let's write

that the LCM which stands for least

common multiple

is 36. so the least common multiple of 9

and 12 is 36. now one thing I do want to

mention about common multiples is that

they are infinite although we only have

one common multiple in our lists as is

36 we can always extend multiples lists

so we can always keep going to find more

common multiples remember multiples are

endless so that means common multiples

are endless so that's just something to

think about when it comes to multiples

let's move on to number two where we

have 10 and 25. let's start with some

multiples

of 10. so 10 times 1 is 10 10 times 2 is

20 10 times 3 is 30 10 times 4 is 40.

and then 10 times 5 is 50. so 10 20 30

40 50 so on and so forth now let's list

the first five multiples

of 25 so 25 times 1 is 25 25 times 2 is

50 25 times 3 is 75 25 times 4 is 100

and then 25 times 5 is 125. now that we

have some multiples listed we can look

for common multiples and specifically

the least common multiple well 50 is a

common multiple

and it happens to be the least common

multiple so the LCM

is

50 the least common multiple of 10 and

25 is 50. so there's how we list out

some multiples of the numbers in order

to find the least common multiple let's

move on to using prime factorization

here are our examples for using prime

factorization now I like using this

strategy and find it helpful when

working with numbers that are a little

larger in value and not as simple to

work with for example the strategy of

listing out multiples of numbers in

order to find the LCM can be kind of

difficult and time consuming when

working with larger numbers in value so

this is a different approach a different

strategy to be familiar with when it

comes to finding the least common

multiple let's jump into our examples

starting with number one where we have

15 and 27. let's start with the prime

factorization of 15 and we will start

with the factors of 3

and five now three is prime so we are

done there

and 5 is prime so we are done there as

well and that's the prime factorization

of 15. we can't break that down any

further now we have

the prime factorization of 27.

let's start with the factors of 3

and 9 3 times 9 equals 27 so 3 and 9 are

factors of 27.

3 is prime so we are done there but we

can break 9 down 3 times 3

equals nine

so three is a factor of nine

three is prime so we are done there

and there and that's the prime

factorization of 27. we can't break that

down any further

now we're ready to move to the next step

so we need to list the prime factors of

15 and 27 and match them vertically

let's see what this looks like starting

with

15. so our prime factors from the prime

factorization are three and five or

three times five

now four

twenty-seven so we have three

times three

times three and you'll notice that big

gap underneath the 5 there we are

matching numbers vertically 27 does not

have a prime factor of five so I left

that blank underneath the 5. now that we

have our prime factors listed and

matched vertically we move on to the

next step where we bring down and I like

to draw a line underneath here in order

to separate these steps so this is a

column

and although we have two threes here

this is a column of Threes so we just

bring

one down we have a three to represent

that column of two threes times

we have a column of five here

times we have a three here

times

another three here

so we end up with three times five times

three times three and by multiplying

these we get our least common multiple

so three times five is fifteen times

three is forty-five times three is one

hundred thirty-five and that's our least

common multiple

so the LCM the least common multiple of

15 and 27

is one hundred thirty-five let's move on

to number two where we have 28 and 52.

let's start with the prime factorization

of 28. now 2 times 14 equals 28 so let's

start with those factors 2 is prime so

we are done there 14 we can break down

two times seven equals 14. so 2 and 7

are factors of 14.

2 is prime so we are done there

and 7 is prime as well so we are done

there and that's the prime factorization

of 28. we can't break that down any

further

now we need the prime factorization of

52. let's start with the factors of 2

and 26 2 times 26 equals 52. so 2 and 26

are factors of 52.

2 is prime so we are done there 26 we

can break that down

2 times 13 equals 26 so 2 and 13 are

factors of 26.

2 is prime so we are done there and 13

is prime as well so we are done there

and that's the prime factorization of

52. we can't break that down any further

now we need to list the prime factors

and match them vertically

428 we have

two

times two

times seven

452

we have two

times two

times

13.

now we need to bring down so we have a

column

of twos here so let's bring down a 2 to

represent that column

times

another column of twos so let's bring

another two down

times

7

times

13.

so we have 2 times 2 times 7 times 13 to

get our least common multiple we have

two times two which is 4 times 7 is 28

times 13. well I'm not sure what 28

times 13 is so let's come to the side

here

and multiply 28 times 13. we will start

with 3 times 8

which is 24 3 times 2 is 6 Plus 2.

is 8. we are done here and done here

we need a zero now we have one times

eight which is eight and then one times

two

is two

let's add

four plus zero is four eight plus eight

is sixteen and then one plus two

is three so we get 364.

so the least common multiple of 28 and

52 let me squeeze this in here is 300

60

4.

so there you have it there's how to find

the least common multiple the LCM using

two different strategies listing out

multiples and using prime factorization

I hope that helped

thanks so much for watching

until next time peace

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