How to Find the LCM using Prime Factorization | Least Common Multiple | Math with Mr. J

Summary

In the video 'Math with Mr J,' the presenter teaches viewers how to find the Least Common Multiple (LCM) using prime factorization, an efficient method for larger numbers. The tutorial begins with the prime factorization of 15 and 27, illustrating the process of breaking down numbers into their prime components. It then matches these factors vertically and multiplies them to find the LCM, which for 15 and 27 is 135. The video continues with another example using 28 and 52, demonstrating the same technique to determine their LCM as 364. This approach simplifies the LCM calculation, making it more manageable for complex numbers.

Takeaways

• 📚 The video teaches how to find the Least Common Multiple (LCM) using prime factorization.
• 🔢 Prime factorization is a strategy suitable for larger numbers that simplifies the LCM calculation.
• 🌰 The example of finding LCM for 15 and 27 is used to demonstrate the process.
• 📝 Prime factorization of 15 is 3 and 5, both of which are prime numbers.
• 📝 Prime factorization of 27 is 3 x 3 x 3, since 9 (3 x 3) is a factor of 27 and 3 is prime.
• 📋 The script instructs to list and match prime factors of the numbers vertically.
• 🔄 For 15 and 27, the matched prime factors are 3 (from 15) and 3 x 3 (from 27), and the unmatched factor is 5 (from 15).
• 🧮 The LCM is calculated by multiplying the highest powers of all prime factors involved, resulting in 135 for 15 and 27.
• 🌰 Another example is given with numbers 28 and 52, showing the step-by-step prime factorization.
• 📝 The prime factorization of 28 is 2 x 2 x 7, and for 52, it's 2 x 2 x 13.
• 📋 The unmatched prime factors for 28 and 52 are 7 and 13, respectively, used to calculate the LCM.
• 🧮 The LCM for 28 and 52 is found by multiplying 2 x 2 x 7 x 13, which equals 364.

Q & A

What is the primary focus of the video by Mr. J?

The primary focus of the video is to teach how to find the least common multiple (LCM) using prime factorization, a method that is particularly useful for larger numbers.

Why is prime factorization a better approach for finding LCM compared to listing multiples?

Prime factorization is a better approach for finding the LCM of larger numbers because listing out multiples can be difficult and time-consuming, whereas prime factorization provides a more systematic and efficient method.

What are the first two numbers Mr. J uses to demonstrate the prime factorization method for finding LCM?

The first two numbers Mr. J uses to demonstrate the method are 15 and 27.

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

The prime factorization of 15 is 3 and 5, as both numbers are prime and cannot be broken down further.

How is the prime factorization of 27 derived in the video?

The prime factorization of 27 is derived by first recognizing that 3 and 9 are factors of 27. Then, since 9 can be broken down further into 3 times 3, the prime factorization is 3 times 3 times 3.

What is the least common multiple of 15 and 27 according to the video?

The least common multiple of 15 and 27, found using prime factorization, is 135.

What is the next pair of numbers Mr. J uses to illustrate the process after 15 and 27?

After demonstrating with 15 and 27, Mr. J uses the numbers 28 and 52 to further illustrate the process of finding LCM using prime factorization.

What is the prime factorization of 28 as explained in the video?

The prime factorization of 28 is 2 times 2 times 7, as 14 (which is 2 times 7) is a factor of 28, and both 2 and 7 are prime numbers.

Can you explain the prime factorization of 52 as presented in the video?

The prime factorization of 52 is 2 times 2 times 13, as 26 (which is 2 times 13) is a factor of 52, and both 2 and 13 are prime numbers.

What is the least common multiple of 28 and 52 as calculated in the video?

The least common multiple of 28 and 52, calculated using prime factorization, is 364.

How does Mr. J suggest multiplying the prime factors to find the LCM in the video?

Mr. J suggests multiplying the highest powers of all prime factors present in the factorization of the given numbers to find the LCM.

Outlines

00:00

📚 Introduction to Finding LCM through Prime Factorization

This paragraph introduces a video tutorial by Mr. J on the method of finding the Least Common Multiple (LCM) using prime factorization. The method is particularly useful for larger numbers that are difficult to handle using the traditional method of listing multiples. The video begins with an example using the numbers 15 and 27. The prime factorization of 15 is identified as 3 and 5, both of which are prime numbers. For 27, the factors are broken down into 3 (a prime number) and 9, which further factors into 3 times 3. The process involves listing the prime factors of both numbers and then matching them vertically to find the LCM. The LCM is calculated by multiplying the highest powers of all prime factors present in either number. For 15 and 27, the LCM is found to be 135 by multiplying the prime factors 3, 5, and 3 (from 27).

05:00

🔢 Continued Explanation of LCM Calculation with Prime Factorization

The second paragraph continues the explanation of finding the LCM using prime factorization with a new example involving the numbers 28 and 52. The prime factorization of 28 is determined to be 2 times 2 times 7, with 2 and 7 being prime numbers. For 52, the factors are 2 times 26, which further breaks down into 2 times 13, with both 2 and 13 being prime. The process involves aligning the prime factors vertically and then multiplying the highest powers of all the prime factors to find the LCM. In this case, the LCM is calculated by multiplying 2 (from 28), another 2 (from 52), 7 (from 28), and 13 (from 52). The multiplication of 28 and 13 is shown step-by-step, resulting in 364. The final LCM of 28 and 52 is 364, which is obtained by multiplying 4 (2 times 2), 28 (from 7 times 4), and 13. The paragraph concludes with a summary of the method and a note of thanks for watching.

Keywords

💡Least Common Multiple (LCM)

The Least Common Multiple (LCM) is the smallest number that is a multiple of two or more numbers. In the context of the video, finding the LCM is the primary goal, and the video explains how to calculate it using prime factorization. The LCM of 15 and 27, for instance, is calculated to be 135, which is the smallest number that both 15 and 27 can divide into without leaving a remainder.

💡Prime Factorization

Prime Factorization is the process of breaking down a number into its smallest prime factors. In the video, prime factorization is used as a method to find the LCM. For example, the number 15 is factored into 3 and 5, both of which are prime numbers, and this process is essential for identifying the prime factors needed to calculate the LCM.

💡Prime Numbers

Prime numbers are numbers greater than 1 that have no divisors other than 1 and themselves. The video emphasizes the importance of identifying prime numbers during prime factorization. For instance, when factoring 27, the number is broken down into 3 (a prime number) and 9, which further factors into 3 (again prime) and 3.

💡Multiples

Multiples are numbers that can be divided by another number without leaving a remainder. The concept of multiples is indirectly referenced in the video when discussing the LCM, as the LCM is essentially the smallest multiple common to all the given numbers. For example, when finding the LCM of 15 and 27, multiples of these numbers are considered until the smallest common multiple is found.

💡Factors

Factors are numbers that divide into another number without leaving a remainder. The video discusses factors in the context of breaking down numbers into their prime components. For example, the number 28 has factors of 2 and 14, which are then further broken down into prime factors of 2, 2, and 7.

💡Divisibility

Divisibility refers to the ability of one number to be divided evenly by another. The concept of divisibility is central to finding the LCM, as it involves determining the smallest number into which all given numbers can be divided without a remainder. The video uses divisibility as a criterion for identifying the LCM of 28 and 52, which is calculated to be 364.

💡Strategy

A strategy in the context of the video refers to a method or plan for solving a problem. The video introduces prime factorization as a strategy for finding the LCM, especially useful for larger numbers that are difficult to handle using the traditional method of listing multiples.

💡Value

In mathematics, 'value' often refers to the numerical amount of a quantity. The video mentions 'value' when discussing the size of numbers, emphasizing that the prime factorization strategy is particularly helpful for numbers with larger values, which can be cumbersome to work with using other methods.

💡Column

In the context of the video, a 'column' refers to a vertical arrangement of the same prime factors when listing them out for LCM calculation. The video instructs to bring down the highest power of each prime factor from the columns to calculate the LCM, such as bringing down two 2s and one 7 from the columns for the numbers 28 and 52.

💡Multiplication

Multiplication is the mathematical operation of adding a number to itself a certain number of times. In the video, multiplication is used to combine the prime factors identified to calculate the LCM. For example, after identifying the prime factors, the video demonstrates multiplying 3, 5, and 3 to get the LCM of 135 for the numbers 15 and 27.

Highlights

Introduction to finding the least common multiple (LCM) using prime factorization.

Explanation of why prime factorization is useful for larger numbers.

Demonstration of prime factorization for the number 15.

Prime factorization of 27, including breaking down the factor 9.

Method of listing prime factors of 15 and 27 vertically for comparison.

Step-by-step process of multiplying the highest powers of prime factors to find LCM.

Calculation of the LCM of 15 and 27, resulting in 135.

Prime factorization of 28, including the factors 2, 2, and 7.

Prime factorization of 52, including the factors 2, 2, and 13.

Matching prime factors of 28 and 52 vertically for LCM calculation.

Multiplication of the highest powers of prime factors for 28 and 52 to find LCM.

Calculation of the LCM of 28 and 52, resulting in 364.

Explanation of the importance of including all prime factors in the LCM calculation.

Emphasis on the efficiency of prime factorization over listing multiples for LCM.

Conclusion and summary of the method for finding LCM using prime factorization.

Encouragement for viewers to practice the method with different numbers.

Closing remarks and sign-off from Mr. J.

Transcripts

welcome to math with Mr J

[Music]

in this video I'm going to cover how to

find the least common multiple also

known as the LCM 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 nine down three times three

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 3 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 3 to represent that

column of two threes

times

we have a column of 5 here

times we have a 3 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 3 times 5 is 15 times 3 is 45 times 3

is

135 and that's our least common multiple

so the LCM the least common multiple of

15 and 27

is

135. 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

2 times 7 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

2

times 2

times 7.

452

we have 2

times 2

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 2 down

times

7

times

13.

so we have 2 times 2 times 7 times 13 to

get our least common multiple we have 2

times 2 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 8 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 using

prime factorization I hope that helped

thanks so much for watching

until next time peace

foreign