Finding the Least Common Multiple of Two Numbers
Learning Outcomes
- Find the least common multiple of two numbers by listing multiples
- Find the least common multiple of two numbers by prime factorization
One of the reasons we find multiples and primes is to use them to find the least common multiple of two numbers. This will be useful when we add and subtract fractions with different denominators.
Listing Multiples Method
A common multiple of two numbers is a number that is a multiple of both numbers. Suppose we want to find common multiples of 10 and 25. We can list the first several multiples of each number. Then we look for multiples that are common to both lists—these are the common multiples.
10 : 10,20,30,40,50,60,70,80,90,100,110…25 : 25,50,75,100,125…
We see that 50 and 100 appear in both lists. They are common multiples of 10 and 25. We would find more common multiples if we continued the list of multiples for each.
The smallest number that is a multiple of two numbers is called the least common multiple (LCM). So the least LCM of 10 and 25 is 50.
Find the least common multiple (LCM) of two numbers by listing multiples
- List the first several multiples of each number.
- Look for multiples common to both lists. If there are no common multiples in the lists, write out additional multiples for each number.
- Look for the smallest number that is common to both lists.
- This number is the LCM.
example
Find the LCM of
15 and
20 by listing multiples.
Solution:
List the first several multiples of
15 and of
20. Identify the first common multiple.
15: 15,30,45,60,75,90,105,12020: 20,40,60,80,100,120,140,160
The smallest number to appear on both lists is
60, so
60 is the least common multiple of
15 and
20.
Notice that
120 is on both lists, too. It is a common multiple, but it is not the least common multiple.
try it
[ohm_question]145458[/ohm_question]
In the next video we show an example of how to find the Least Common Multiple by listing multiples of each number.
https://youtu.be/7twRSmgcrLM
Prime Factors Method
Another way to find the least common multiple of two numbers is to use their prime factors. We’ll use this method to find the LCM of 12 and 18.
We start by finding the prime factorization of each number.
12=2⋅2⋅318=2⋅3⋅3
Then we write each number as a product of primes, matching primes vertically when possible.
12=2⋅2⋅318=2⋅3⋅3
Now we bring down the primes in each column. The LCM is the product of these factors.
Notice that the prime factors of 12 and the prime factors of 18 are included in the LCM. By matching up the common primes, each common prime factor is used only once. This ensures that 36 is the least common multiple.
Find the LCM using the prime factors method
- Find the prime factorization of each number.
- Write each number as a product of primes, matching primes vertically when possible.
- Bring down the primes in each column.
- Multiply the factors to get the LCM.
example
Find the LCM of
15 and
18 using the prime factors method.
Answer:
Solution:
Write each number as a product of primes. |
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Write each number as a product of primes, matching primes vertically when possible. |
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Bring down the primes in each column. |
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Multiply the factors to get the LCM. |
LCM=2⋅3⋅3⋅5The LCM of 15 and 18 is 90. |
try it
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example
Find the LCM of
50 and
100 using the prime factors method.
Answer:
Solution:
Write the prime factorization of each number. |
50=2⋅5⋅5100=2⋅2⋅5⋅5 |
Write each number as a product of primes, matching primes vertically when possible. |
50=2⋅5⋅5100=2⋅2⋅5⋅5 |
Bring down the primes in each column. |
 |
Multiply the factors to get the LCM. |
LCM=2⋅2⋅5⋅5The LCM of 50 and 100 is 100. |
try it
[ohm_question]145462[/ohm_question]
In the next video we show how to find the Least Common Multiple by using prime factorization.
https://youtu.be/hZvRDG-HgMYLicenses & Attributions
CC licensed content, Original
- Question ID: 145462, 145459, 145458. Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
- Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
CC licensed content, Shared previously
- Example: Determining the Least Common Multiple Using a List of Multiples. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
- Example: Determining the Least Common Multiple Using Prime Factorization. Authored by: James Sousa (Mathispower4u.com). License: CC BY: Attribution.
CC licensed content, Specific attribution
- Prealgebra. Provided by: OpenStax License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757.