How to Find the LCM using Prime Factorization | Least Common Multiple | Math with Mr. J
Summary
TLDRIn 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
- ๐ The video teaches how to find the Least Common Multiple (LCM) using prime factorization.
- ๐ข Prime factorization is a strategy suitable for larger numbers that simplifies the LCM calculation.
- ๐ฐ The example of finding LCM for 15 and 27 is used to demonstrate the process.
- ๐ Prime factorization of 15 is 3 and 5, both of which are prime numbers.
- ๐ Prime factorization of 27 is 3 x 3 x 3, since 9 (3 x 3) is a factor of 27 and 3 is prime.
- ๐ The script instructs to list and match prime factors of the numbers vertically.
- ๐ 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).
- ๐งฎ The LCM is calculated by multiplying the highest powers of all prime factors involved, resulting in 135 for 15 and 27.
- ๐ฐ Another example is given with numbers 28 and 52, showing the step-by-step prime factorization.
- ๐ The prime factorization of 28 is 2 x 2 x 7, and for 52, it's 2 x 2 x 13.
- ๐ The unmatched prime factors for 28 and 52 are 7 and 13, respectively, used to calculate the LCM.
- ๐งฎ 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
๐ 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).
๐ข 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.
Mindmap
Keywords
๐กLeast Common Multiple (LCM)
๐กPrime Factorization
๐กPrime Numbers
๐กMultiples
๐กFactors
๐กDivisibility
๐กStrategy
๐กValue
๐กColumn
๐กMultiplication
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
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