Lesson Objectives

- Learn the definition of a multiple
- Learn how to generate a list of multiples for a number
- Learn how to find the LCM using the listing method
- Learn how to build the LCM using the prime factorization of each number

## How to Find the Least Common Multiple (LCM)

In a previous lesson, we learned how to find the GCF or greatest common factor for a group of numbers. We learned that the GCF was the
largest number that was a factor or divisor for the group. In this lesson, we will focus on how to find the LCM or least common multiple. The LCM is the smallest
number that is a multiple of each number of the group.

4 x 1 = 4

4 x 2 = 8

4 x 3 = 12

4 x 4 = 16

4 x 5 = 20

In other words, we could start with 4 and increase by 4 to get to the next multiple:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40,…

We know the "..." is an ellipsis and indicates the pattern continues forever. A number will have an infinite (unlimited) number of multiples, but a finite (limited) number of factors. The number 4 has only three factors: 1, 2, and 4.

Example 1: Show the multiples of 11:

11, 22, 33, 44, 55, 66, 77, 88, 99, 110,...

2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...

3: 3, 6, 9, 12, 15, 18, 21,...

We can see that 6, 12, and 18 are common multiples of 2 and 3. The smallest or least is 6. The least common multiple of 2 and 3, shown as LCM(2, 3) is 6. Finding the LCM by listing the multiples of each number is very inefficient. This is just done to give an explanation of the concept. Let's look at one more example using the listing approach, and then we will move to a more efficient method.

Example 2: Find the LCM of 4 and 6 using the listing method:

4: 4, 8, 12, 16,...

6: 6, 12, 18,...

We can see that our least common multiple of 4 and 6 is 12.

LCM(4, 6) = 12

Building the LCM

Example 3: Find the LCM of 21 and 45

Let's start with the prime factorization of each number:

21 = 3 x 7

45 = 3 x 3 x 5

We can use a table to help organize information. This isn't necessary, but is helpful when we first solve these types of problems:

We can see the prime factors involved are: 3, 5, and 7. Each of these numbers will go into our list. The key is understanding what to do about a prime
factor that appears in more than one factorization. We only include the largest number of repeats from any of the prime factorizations.
In this case, 3 occurs in the prime factorization of both 21 and 45. In the prime factorization of 45, there are two factors of 3. In the prime factorization of 21,
there is only one. The largest number of repeats is two, so two factors of 3 go into our list.

LCM List: 3, 3, 5, 7

Now we can multiply the numbers on the list to find the LCM

LCM(21, 45) = 3 x 3 x 5 x 7 = 315

What if we asked for the GCF of 21 and 45:

From our table, we can see that only one factor of 3 is common to all. This means the GCF will be 3.

GCF(21, 45) = 3

It can be very easy to confuse the GCF and LCM. We start each procedure by finding the prime factorization of all numbers in the group. When we look for the GCF, we want the largest factor or divisor for a group of numbers. This means we are searching for prime factors that are common to all. If a factor is not common to every number of the group, it does not go into the list. When we look for the LCM, we want the smallest multiple that is common to a group of numbers. This means every prime factor from all numbers will go into the list, whether they are common or not. The only exception is when a factor is common; in this case, we only include the largest number of repeats between any of the factorizations. It can be counterintuitive, but with the GCF think (factor, divisor) - smaller and with LCM think (multiple) - larger.

Let's try another example.

Example 4: Find the LCM of 52 and 88

Let's start with the prime factorization of each number:

52 = 2 x 2 x 13

88 = 2 x 2 x 2 x 11

We can use a table to help organize information. This isn't necessary, but is helpful when we first solve these types of problems:

We can see the prime factors involved are: 2, 11, and 13. Each of these numbers will go into our list. The key is understanding what to do about a prime
factor that appears in more than one factorization. We only include the largest number of repeats from any of the prime factorizations.
In this case, 2 occurs in the prime factorization of both 52 and 88. In the prime factorization of 88, there are three factors of 2. In 52, there are only two. The
largest number of repeats is three, so three factors of 2 go into our list.

LCM List: 2, 2, 2, 11, 13

Now we can multiply the numbers on the list to find the LCM

LCM(52, 88) = 2 x 2 x 2 x 11 x 13 = 1144

What if we asked for the GCF of 52 and 88:

From our table, we can see that two factors of 2 are common to all. This means the GCF will be 4.

GCF(52, 88) = 4

Let's take a look at an example with three numbers involved.

Example 5: Find the LCM of 6, 18, and 105

Let's start with the prime factorization of each number:

6 = 2 x 3

18 = 2 x 3 x 3

105 = 3 x 5 x 7

We can use a table to help organize information. This isn't necessary, but is helpful when we first solve these types of problems:

We can see the prime factors involved are: 2, 3, 5 and 7. Each of these numbers will go into our list. The key is understanding what to do about a prime
factor that appears in more than one factorization. We only include the largest number of repeats from any of the prime factorizations.
In this case, 2 occurs in the prime factorization of both 6 and 18. In the prime factorization of both 6 and 18, there is one factor of 2. This means the largest number
of repeats is one, so one factor of 2 will go into our list. Additionally, we have a 3 that occurs in the prime factorization of each number. 3 appears only once in the prime
factorization of both 6 and 105, but twice in the prime factorization of 18. The largest number of repeats is two, so two factors of 3 will go into the list.

LCM List: 2, 3, 3, 5, 7

Now we can multiply the numbers on the list to find the LCM

LCM(6, 18, 105) = 2 x 3 x 3 x 5 x 7 = 630

What if we asked for the GCF of 6, 18 and 105:

From our table, we can see that only one factor of 3 is common to all. This means the GCF will be 3.

GCF(6, 18, 105) = 3

### What is a Multiple?

What exactly is a multiple? A multiple of a number is the product of the number and an integer. For the purposes of simplicity, we will restrict our definition in this course to: a multiple of a number is the product of the number and a nonzero whole number. In other words, the multiples of a number are found by multiplying the number by each nonzero whole number. Let’s take look at a few multiples of 4:4 x 1 = 4

4 x 2 = 8

4 x 3 = 12

4 x 4 = 16

4 x 5 = 20

In other words, we could start with 4 and increase by 4 to get to the next multiple:

4, 8, 12, 16, 20, 24, 28, 32, 36, 40,…

We know the "..." is an ellipsis and indicates the pattern continues forever. A number will have an infinite (unlimited) number of multiples, but a finite (limited) number of factors. The number 4 has only three factors: 1, 2, and 4.

Example 1: Show the multiples of 11:

11, 22, 33, 44, 55, 66, 77, 88, 99, 110,...

### What are Common Multiples?

Common multiples are multiples that are common to all numbers in a group. Let's take a look at the multiples for 2 and 3:2: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...

3: 3, 6, 9, 12, 15, 18, 21,...

We can see that 6, 12, and 18 are common multiples of 2 and 3. The smallest or least is 6. The least common multiple of 2 and 3, shown as LCM(2, 3) is 6. Finding the LCM by listing the multiples of each number is very inefficient. This is just done to give an explanation of the concept. Let's look at one more example using the listing approach, and then we will move to a more efficient method.

Example 2: Find the LCM of 4 and 6 using the listing method:

4: 4, 8, 12, 16,...

6: 6, 12, 18,...

We can see that our least common multiple of 4 and 6 is 12.

LCM(4, 6) = 12

### How to Find the Least Common Multiple using Prime Factorizations

Instead of listing the multiples of each number and looking for the smallest common multiple, we can build the LCM from the prime factorization of each number.Building the LCM

- Find the prime factorization of each number
- Use a table or any organization method to list the prime factors of each number
- Generate a list that contains each prime factor. When a factor is repeated between two or more numbers, we only include the largest number of repeats from any of the prime factorizations
- The LCM is the product of the numbers in the list

Example 3: Find the LCM of 21 and 45

Let's start with the prime factorization of each number:

21 = 3 x 7

45 = 3 x 3 x 5

We can use a table to help organize information. This isn't necessary, but is helpful when we first solve these types of problems:

Number | Prime Factors | ||||
---|---|---|---|---|---|

21 | 3 | 7 | |||

45 | 3 | 3 | 5 |

LCM List: 3, 3, 5, 7

Now we can multiply the numbers on the list to find the LCM

LCM(21, 45) = 3 x 3 x 5 x 7 = 315

What if we asked for the GCF of 21 and 45:

From our table, we can see that only one factor of 3 is common to all. This means the GCF will be 3.

GCF(21, 45) = 3

It can be very easy to confuse the GCF and LCM. We start each procedure by finding the prime factorization of all numbers in the group. When we look for the GCF, we want the largest factor or divisor for a group of numbers. This means we are searching for prime factors that are common to all. If a factor is not common to every number of the group, it does not go into the list. When we look for the LCM, we want the smallest multiple that is common to a group of numbers. This means every prime factor from all numbers will go into the list, whether they are common or not. The only exception is when a factor is common; in this case, we only include the largest number of repeats between any of the factorizations. It can be counterintuitive, but with the GCF think (factor, divisor) - smaller and with LCM think (multiple) - larger.

Let's try another example.

Example 4: Find the LCM of 52 and 88

Let's start with the prime factorization of each number:

52 = 2 x 2 x 13

88 = 2 x 2 x 2 x 11

We can use a table to help organize information. This isn't necessary, but is helpful when we first solve these types of problems:

Number | Prime Factors | ||||
---|---|---|---|---|---|

52 | 2 | 2 | 13 | ||

88 | 2 | 2 | 2 | 11 |

LCM List: 2, 2, 2, 11, 13

Now we can multiply the numbers on the list to find the LCM

LCM(52, 88) = 2 x 2 x 2 x 11 x 13 = 1144

What if we asked for the GCF of 52 and 88:

From our table, we can see that two factors of 2 are common to all. This means the GCF will be 4.

GCF(52, 88) = 4

Let's take a look at an example with three numbers involved.

Example 5: Find the LCM of 6, 18, and 105

Let's start with the prime factorization of each number:

6 = 2 x 3

18 = 2 x 3 x 3

105 = 3 x 5 x 7

We can use a table to help organize information. This isn't necessary, but is helpful when we first solve these types of problems:

Number | Prime Factors | ||||
---|---|---|---|---|---|

6 | 2 | 3 | |||

18 | 2 | 3 | 3 | ||

105 | 3 | 5 | 7 |

LCM List: 2, 3, 3, 5, 7

Now we can multiply the numbers on the list to find the LCM

LCM(6, 18, 105) = 2 x 3 x 3 x 5 x 7 = 630

What if we asked for the GCF of 6, 18 and 105:

From our table, we can see that only one factor of 3 is common to all. This means the GCF will be 3.

GCF(6, 18, 105) = 3

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