2. Greatest Common Factor and Least Common Multiple
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Chapter 1
2.
Greatest Common Factor and Least Common Multiple
This lesson delves into the mathematical concepts of greatest common factor (GCF) and least common multiple (LCM). It explains how to find these values using prime factorization and factor trees. Understanding GCF and LCM is crucial for various applications, from simplifying fractions in basic arithmetic to solving complex problems in engineering and computer science. The lesson uses factor trees to break down numbers into their prime factors, making it easier to find the GCF and LCM.
Show less Show more expand_more| Student Learning Objectives: |
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| | 16 Theory slides |
| | 16 Exercises - Grade E - A |
| | Each lesson is meant to take 1-2 classroom sessions |
Greatest Common Factor and Least Common Multiple
Slide of 16
Think about a delicious hot dog for a snack. A familiar problem for people who prepare it is knowing how many packs of hot dogs and buns are needed to have no leftovers when hot dogs typically come in packs of 10 and buns come in packs of 8. Although it may sound challenging, we can use the greatest common factor and least common multiple to solve this and other similar situations. These properties will be discussed in this lesson.
Catch-Up and Review
Here are a few recommended readings before getting started with this lesson.
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Comparing Factors of 12 and 18
The following applet creates arrangements using the factors of 12 and 18 as the width of an arrangement of blocks. It considers vertical and horizontal arrangements as different arrangements.
Now consider the following questions.
- How many factors do these two numbers have in common?
- What is the greatest factor of both 12 and 18 that arranges the blocks into rectangles with the same width?
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Discussion
Breaking a Number Into Prime Factors
An essential property of any whole number greater than 1 is that it can be expressed as a product of prime numbers. This process is called prime factorization.
Concept
Prime Factorization
Prime factorization, also called complete factorization, is the decomposition of a whole number into a product of its prime factors. 12 &= 2*2*3 &⇕ 12 &= 2^2* 3 The prime factors are found by dividing out the number by the smallest prime number that is a factor of that number. This process is repeated with the quotient until the resulting quotient is a prime number. The following table shows the prime factorization of 80.
| Number | Smallest Prime Factor | Quotient |
|---|---|---|
| 80 | 2 | 80/2 = 40 |
| 40 | 2 | 40/2 = 20 |
| 20 | 2 | 20/2 = 10 |
| 10 | 2 | 10/2 = 5 |
| Prime Factorization | ||
| 80 = 2* 2* 2* 2* 5 | ||
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Discussion
Factor Tree
A factor tree is a diagram that shows the prime factors of a number. The tree begins with a root node that contains the number whose prime factorization is needed. Two branches extend from the root node and connect to a factor pair of the number. This process continues by breaking each factor into its factors until only prime factors appear at the end of the branches.
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Example
Use Factor Trees to Find the Missing Values in a Math Puzzle
Magdalena loves puzzles. She is currently solving a special edition of a sudoku, a game that consists of a 9* 9 grid with some cells containing numbers and others blank. The purpose of the game is to find the missing values by using the numbers 1-9 only once in each row and column and in each of the nine 3*3 boxes.
Magdalena is filling in the red and yellow cells.
a
The number that goes in the red cell is the most repeated prime factor of the prime factorization of 54. What value goes in the red cell?
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b
The yellow cell contains the most repeated factor of the prime factorization of 64. What value goes in the yellow cell?
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Hint
a
Make a factor tree of 54.
b
Make a factor tree of 64.
Solution
a
To find the missing value on the red cell, we will find the prime factorization of 54. Let's make a factor tree for this purpose.
The missing value in the red cell is the most common prime factor in the prime factorization of 54. 54=2* 3* 3* 3 Here, the number 3 is repeated three times, while the number 2 appears only once. Therefore, the value in the red cell is 3.
b
Let's find the prime factorization of 64 in a similar way as we did in Part A.
The prime factorization of 64 contains only the number 2 six times. 64=2*2*2*2*2*2 Therefore, the missing value in the yellow cell of the sudoku game is 2. Magdalena can now fill in the yellow and red cells in the sudoku board!
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Discussion
Greatest Common Factor
Factors are used to divide a number into equal amounts. However, it may be difficult when two or more different numbers are to be divided into a certain number of groups with equal amounts of each item. The common factors of the numbers, and the greatest common factor in particular, can help find a solution to this type of problem.
Concept
Greatest Common Factor
Factors that are shared by two or more numbers are called common factors. The greatest of these common factors is called the greatest common factor (GCF). Consider, for example, the factors of 4 and 8. Factors of4: & 1, 2, 4 Factors of8: & 1, 2, 4,8 The common factors of 4 and 8 are 1, 2, and 4. The GCF(4,8)= 4. The GCF can also be determined by multiplying the prime factors shared in the prime factorization of the numbers in question.
greatest common divisorbecause a factor of a number divides that number evenly.
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Discussion
Finding the Greatest Common Factor
The greatest common factor (GCF) of two numbers can be determined by finding the prime factorization of the numbers. The GCF is the product of the common prime factors. Consider the following pair of numbers. 42,24 The GCF(42,24) will be found by following these four steps.
1
Find the Prime Factorization of the First Number
expand_more 
2
Find the Prime Factorization of the Second Number
expand_more Now, follow the same process to find the prime factorization of the second number.
3
Identify the Common Prime Factors
expand_more Write the two prime factorizations together and circle the common prime factors. In this example, the prime factorizations of 42 and 24 will be written together.
42=& 2* 3 * 7 24=& 2* 2 * 2 * 3
4
Multiply the Prime Common Factors
expand_more The greatest common factor of the numbers is the product of the common prime factors of the prime factorization. In this case, the common prime factors of 42 and 24 are 2 and 3.
2* 3= 6
Therefore, the greatest common factor of 42 and 24 is 6.
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Example
Find the GCF to Fill in a Set of Sudoku Cells
Magdalena continues with the process of solving her sudoku game. This time, she wants to know which number goes in the purple cells.
The purple cells must be filled with the greatest common factor (GCF) of 35 and 84. What number goes into the purple cells?
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Hint
Find the prime factorizations of 35 and 84. Identify the common prime factors between the two prime factorizations. Multiply the common prime factors to get the GCF.
Solution
The clue claims that the number in the purple cells is the greatest common factor of 35 and 85. To find the GCF, let's first find the prime factorization of the two numbers.
Prime Factorization of 35
We will find the prime factorization of 35 using a factor tree.
The prime factorization of 35 is 5* 7.
Prime Factorization of 84
We will follow the same process to find the prime factorization of 84.
The prime factorization of 84 is 2* 2* 3* 7.
Identify Common Prime Factors
Now that we have the two prime factorizations, let's find the common prime factor shared by these factorizations. &35=5* 7 &84=2*2*3* 7 The only common prime factor shared by these factorization is the number 7.
Multiply
Since only one factor is shared by the prime factorizations of 35 and 84, the greatest common factor of 35 and 84 is 7. This number goes into the purple cells.
What a great achievement! Magdalena has made great progress on her puzzle.
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Comparing Multiples of 12 and 18
Consider the following applet that creates different arrangements of blocks using multiples of 12 and 18.
Consider the following questions.
- Do the numbers have any multiples in common?
- What is the least of the common multiples?
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Discussion
Least Common Multiple
Similar to factors, it may be of interest to find the smallest multiple of two or more different numbers. This number is called the least common multiple.
Concept
Least Common Multiple
The least common multiple (LCM) of two whole numbers a and b is the smallest whole number that is a multiple of both a and b. It is denoted as LCM(a,b). The least common multiple of a and b is the smallest whole number that is divisible by both a and b.
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Discussion
Finding the Least Common Multiple
To determine the least common multiple (LCM) of two or more numbers, begin by finding the prime factorization of each number. Then, highlight all the instances of each prime factor that is repeated most. Finally, multiply the highlighted factors to find the LCM. The LCM(54,60) will be found by following these four steps.
1
Find the Prime Factorization of the First Number
expand_more Use a factor tree to determine the prime factorization of each number. For this situation, the factor tree of 54 will be drawn. 
2
Find the Prime Factorization of the Second Number
expand_more Follow the same process to find the prime factorization of the next number. Consider the factor tree of 60.
3
Match Prime Factors Vertically
expand_more It can be helpful to write the prime factorizations of the numbers in a table. Place each prime factorization in a row of the table. Use columns for each factor and match common factors vertically when possible. The table for the prime factorization of 54 and 60 is shown here.
4
Bring Down the Primes in Each Column and Multiply
expand_more Lastly, bring down the prime factors in each column of the table. This process is done with the table containing the prime factorizations of 54 and 60 below.
The product of these prime factors is the least common multiple of the numbers.
| Prime Factors | Multiplication | LCM(54,60) |
|---|---|---|
| 2, 2, 3, 3, 3 and 5 | 2*2*3*3*3*5 | 540 |
The least common multiple of 54 and 60, which can also be written as LCM(54,60), is 540.
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Example
Cracking a Pair of Mystery Numbers With the LCM
Magdalena is having a great time solving her special Sudoku puzzle but the next clue looks scary.
Magdalena first must find the least common multiple (LCM) of 10 and 16. Next, these conditions must be met.
- The number for the orange cells multiplied by 10 is the LCM(10,16).
- The number for the pink cells multiplied by 16 is the LCM(10,16).
What are the numbers that Magdalena will write in the orange and pink cells? Write the number for the orange cells first.
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Hint
Find the prime factorization of each number. Highlight each factor that is repeated most. Multiply the highlighted factors.
Solution
We first need to find the least common multiple (LCM) of 10 and 16. Let's use factor trees to find each prime factorization.
Match Prime Factors Vertically
Now we will write the prime factorizations of 10 and 16 in a table. The factorization of 10 will be in the first row and the factorization of 16 will be in the second row. Let's line up matching factors when possible.
| 10 | 2 | 5 | |||
|---|---|---|---|---|---|
| 16 | 2 | 2 | 2 | 2 |
Bring Down the Primes in Each Column and Multiply
Lastly, let's bring down the factors of 10 and 16 in each column of the table.
| 10 | 2 | 5 | |||
|---|---|---|---|---|---|
| 16 | 2 | 2 | 2 | 2 | |
| Factors | 2 | 2 | 2 | 2 | 5 |
The product of these prime factors is the least common multiple of 10 and 16.
| Factors | Multiplication | LCM(10,16) |
|---|---|---|
| 2, 2, 2, 2, and 5 | 2*2*2*2*5 | 80 |
Finding the Numbers to Fill in the Orange and Pink Cells
The LCM(10,16) is 80 but we are not ready to fill in the cells in the puzzle yet. For the orange cells, we need to find the number that gives 80 when multiplied by 10. orange* 10=80 ⇒ orange= 8 Let's use the same idea to find the number that goes in the pink cells. pink*16=80 ⇒ pink= 5 The number that goes in the orange cells is 8 and the number that goes in the pink cells is 5. What a big step Magdalena has made now in her solution process!
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Pop Quiz
The GCF and the LCM of Random Numbers
Find the greatest common factor (GCF) or the least common multiple (LCM) of the given numbers, as requested.
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Discussion
Distributive Property
The greatest common factor can help us simplify numeric expressions. Consider the following sum. 48+30 The greatest common factor of these numbers is 6. This means that each number can be rewritten using this common factor. 48+30 ⇔ 6* 8+ 6* 5 The greatest common factor can be pulled out of each addend. This factor will be multiplied by the sum of the numbers left after pulling it out. 6* 8+ 6* 5 ⇔ 6( 8+ 5) The property applied to pull the greatest common factor out of a sum is called the Distributive Property. This property states that multiplying a number by the sum of two or more addends produces the same result as multiplying the number by each addend individually and then adding all the products together.
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Example
Creating Different Bouquets With the Same Amount of Flowers
Magdalena's mother works as a volunteer in a retirement home. She plans to gift some bouquets to the ladies in the retirement home next weekend.
She has 49 roses and 84 tulips. Each bouquet will have the same number of flowers and contain only roses or only tulips.
a
What is the greatest number of flowers that Magdalena's mother can use in each bouquet?
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b
How many bouquets can Magdalena's mother create?
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Hint
a
Find the greatest common factor of 49 and 89.
b
Write an expression for the total number of flowers. Use the Distributive Property to rewrite the expression for the total number of flowers.
Solution
a
Magdalena's mother wants each bouquet to have an equal number of flowers. This number must be the greatest possible number of flowers per bouquet. Also, each bouquet must contain only one type of flower. In other words, they need to find the greatest common factor of 49 and 89.
GCF(49,89)=? First, let's find the prime factorizations of these numbers using a factor tree.
Now, let's look for the common factors between the prime factorizations. &49=7* 7 &84 = 2*2*3* 7 The only common factor between the factorizations is 7, so the GCF(49,84) is 7. Therefore, each bouquet will contain exactly 7 flowers!
b
We will start by finding the total number of flowers available for the bouquets.
49 roses + 84 tulips= 133 flowers In Part A, we found that Magdalena's mother can create bouquets that have 7 flowers each. So, to get the total number of bouquets, let's divide the total number of flowers by 7. 133/7 = 19 Magdalena's mother can create 19 bouquets in total.
Checking Our Answer
Checking if the Number of Bouquets for Each Flower Is CorrectIn Part B, we found that Magdalena's mother can make 19 bouquets from 133 flowers. We know that she creates bouquets that can contain only one type of flower. Let's make sure that she truly can make 19 bouquets in this way.
Number of Rose Bouquets
To find the number of rose bouquets she can create, let's divide the total number of roses, 49, by the number of flowers per bouquet, 7. 49/7 = 7 She can make 7 rose bouquets.
Number of Tulip Bouquets
We also know that there are 84 tulips. Let's do a similar calculation to find the number of tulip bouquets. 84/7 = 12
Total Number of Bouquets
Magdalena's mother can create 7 rose bouquets and 12 tulip bouquets. Overall, she can create 7 + 12 = 19 bouquets. Therefore, our answer of 19 bouquets is correct!
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Closure
Completing the Puzzle
Magdalena's sudoku board is almost done.
Solve the following situations to help Magdalena complete that challenging puzzle.
a
The blue cells must be filled in with the solution to this riddle.
|
A citywide high school soccer club has 100 juniors, 88 sophomores, and 76 seniors. The head coach wants to divide the students into groups of the same size. Each team must have the same numbers of juniors, sophomores, and seniors. What is the greatest possible number of groups the coach can make? |
What is the number to fill in the blue cells?
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b
The green cells must be filled in with the solution to this riddle.
|
Davontay, Vincenzo, and Tadeo saw each other at the movies today. If Davontay goes to the movies every 6th day, Vincenzo every 10th day, and Tadeo every 5th day, how many times must Tadeo go before the three friends will meet at the movies again? |
What is the number to fill in the green cells?
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Solution
a
Let's recall the condition the groups have to satisfy.
|
Each group must contain the same number of juniors, sophomores, and seniors. |
There are 100 juniors, 88 sophomores, and 76 seniors in the club. Since the coach wants to divide each grade level evenly to create the greatest number of groups possible, he wants to find the greatest common factor of these numbers. Let's list the factors of each number.
| Number | Factors |
|---|---|
| 100 | 1, 2, 4, 5, 10, 20, 25, 50, 100 |
| 88 | 1, 2, 4, 8, 11, 22, 44, 88 |
| 76 | 1, 2, 4, 19, 38, 76 |
Notice that these numbers have 1, 2, and 4 as common factors, so 4 is the greatest common factor. This means that the greatest number of groups that the coach can make is 4. Therefore, the number for the blue cells of Magdalena's puzzle is 4.
b
The riddle says that Davontay goes to the movies every 6th day, Vincenzo every 10th day, and Tadeo every 5th day. This means that Davontay's visits are on multiples of 6, Vincenzo's on multiples of 10, and Tadeo's on multiples of 5. Let's list these multiples.
| Number | Multiples |
|---|---|
| 6 | 6, 12, 18, 24, 30, 36, 42, ... |
| 10 | 10, 20, 30, 40, 50, 60, 70, ... |
| 5 | 5, 10, 15, 20, 25, 30, 35, ... |
The least common multiple of 6, 10, and 5 is 30, so it will take 30 days for the boys to visit the movies on the same day again. Since Tadeo goes to the movies every 5 days, 5 times a certain number must equal 30. 5* ? &= 30 ? &= 6 This number is 6, so Tadeo will go to the movies 6 times before meeting his friends again. This is the solution to the riddle, and Magdalena can fill in the green cells with the number 6.
Great news — the remaining numbers are easy to decipher and Magdalena has now filled in the rest of the board. What an outstanding achievement!
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Greatest Common Factor and Least Common Multiple
Exercise 2.1
He wants to create as many groups as possible.
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The choir consists of 80 sixth-grade students and 96 seventh-grade students. 80 students in sixth grade 96 students in seventh grade We want to form groups with the same number of students in each grade. This means the number of students in each grade must be divided evenly by the total number of groups. This means that we can find the possible numbers of groups by finding the factors of 80 and 96. Let's list these factors! Factors of80: & 1, 2, 4, 5, 8, 10, 16, &20, 40, 80 Factors of96: & 1, 2, 3, 4, 6, 8, 12, & 16, 24, 32, 48, 96 The common factors of these numbers are the numbers of possible groups the music teacher could form. Number of Possible Groups: 1, 2, 4, 8, 16 The greatest of these numbers, 16, is the greatest common factor of 80 and 96. This also represents the greater number of groups the teacher could create.
We found previously that the greatest number possible is 16. Let's a be the number of sixth-grade students and b the number of students of seventh-grade that form each group. Note that the sum of a and b represents the total number of students each group will have.
Total Students per Group:
a+ b
Multiplying this sum by the number of groups will result in the total number of students in the choir.
Total Number of Students in the Choir:
16( a+ b)
We can now apply the Distributive Property to this expression.
16( a+ b)
⇕
16* a+ 16* b
The product of 16 and a will equal 80 because there are 80 sixth-grade students.
Number of Sixth-Grade Students Per Group:
16* a=80
⇓
a= 5
We can follow a similar procedure to calculate the number of seventh-grade students per group.
Number of Seventh-Grade Students Per Group:
16* b=96
⇓
b= 6
This means that there will be six seventh-grade students in each group.
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Solution
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We are told that Tiffaniqua has 26 blue marbles and 39 red marbles. Blue Marbles: 26 Red Marbles: 39 Tiffaniqua wants to split these marbles into equal groups. Each group will contain the same number of red marbles and the same number of blue marbles without any marbles left over. The possible groups are given by the common factors of 26 and 39. Let's list the factors of each! Factors of26: & 1, 2, 13, 26 Factors of39: & 1, 3, 13, 39 The common factors are 1 and 13. The greater of these common factors is 13. This means that the greatest common factor of 26 and 39 is 13. The highest number of friends Tiffaniqua could give a bag to without having any leftover marbles is 13.
We found that the highest number of friends Tiffaniqua could give bags to is 13. She has 26 blue marbles. We want to find the number of blue marbles that she should put in each bag. We can do this by finding what number times 13 is 26. Let's let b be this number. 13* b &= 26 &⇓ b&= 2 This means that each of the 13 bags Tiffaniqua will give her friends should contain two blue marbles.
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Davontay is helping his sister make ribbon ornaments for their Christmas tree. They have three ribbons to make these ornaments. The ribbons are green, blue, and orange.
The green ribbon is 96 inches long, the blue one 72 inches, and the orange 60 inches. Davontay's sister asks him to cut the ribbons into strips of equal length and to make sure they are as long as possible. How long should Davontay cut the ribbons to be?
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We are told that the siblings have three ribbons of different lengths. The green ribbon is 96 inches long, 72 inches for the blue one, and 60 inches for the orange ribbon. These ribbons must be divided into equal lengths while keeping them as long as possible.
This means we should find the greatest number that divides these three measures evenly, without any remainder. We can do this by finding the greatest common factor of 96, 72, and 60. Let's list the factors of these three numbers.
| Factors | |
|---|---|
| 96 | 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96 |
| 72 | 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 |
| 60 | 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 |
We can see from the table that the greatest common factors is 12. This means that Davontay will cut each ribbon into 12-inch pieces. We can also find how many pieces he can cut from each ribbon if we divide their lengths by 12. Let's do it!
| Size (Inches) | Number of Strips Produced | |
|---|---|---|
| Green | 96 | 96/12=8 |
| Blue | 72 | 72/12=6 |
| Orange | 60 | 60/12=5 |
This was a big step in the decoration process. Davontay and her sister created the most beautiful Christmas tree they had ever made.
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At the supermarket, they notice that hot dogs come in packs of 10, while buns come in packs of eight. Zain's mother says that she has to make 80 hot dogs to have no hot dogs or buns left over.
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We want find the smallest number divisible by the numbers of hot dogs and buns in each pack. This means we should find the least common multiple (LCM) of 10 and 8. This will help us determine whether Zain's mother is right. Let's list the first multiples of 10 and 8. Multiples of10:& 10, 20, 30, 40, 50, 60, & 70, 80, 90, 100, ... Multiples of8:& 8, 16, 24, 32, 40, 48, & 56, 64, 72, 80, ... We can see from these lists that two common multiples of 10 and 8 are 40 and 80. The LCM( 10, 8) is 40. This means that Zain's mother is not right. The least amount of hot dogs she can make and not have any hot dogs or buns left over is 40.
Let's now find the smallest number of packs of buns and hot dogs that Zain's mother must buy and not have any leftovers. We know that hot dogs come in packs of 10 and buns in packs of 8. We also know that the LCM( 10, 8)= 40. Let's divide 40 by 10 and 8 to find how many packs of each are needed. Packs of Hot Dogs: 40/10=4 packs [0.8em] Packs of Buns: 40/8=5 packs This means that Zain's mother should buy 4 packs of hot dogs and five packs of buns in order to make the smallest number of hot dogs without any leftovers.
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Ali is helping his grandma plant sunflowers in her garden.
They have enough sunflowers to plant rows of either 5 or 6 flowers if every row has the same number of flowers. What is the least number of sunflowers they could have?
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We are told that Ali and his grandma have enough sunflowers to plant 5 or 6 in each row of a garden.
This means that the total number of sunflowers they have is a multiple of both 5 and 6. Let's list the first multiples of 5 and 6 to give us an idea of the possible number of sunflowers they could have. Multiples of5:& 5, 10, 15, 20, 25, 30, 35, 40, & 45, 50, 55, 60, ... Multiples of6:& 6, 12, 18, 24, 30, 36, 42, 48, & 54, 60, 66, 72, ... Two possible options are 30 and 60. The least common multiple LCM of 5 and 6 is 30, so the least number of sunflowers Ali and his grandma could have is 30.
We can also use the prime factorization to find this number. Let's look at the prime factorizations of 5 and 6.
| Number | Prime Factorization |
|---|---|
| 5 | 5 |
| 6 | 2*3 |
We can see that we have three different factors in the prime factorizations. Since these prime factors only appear once, their product will give us the LCM( 5, 6). LCM( 5, 6)=2*3*5 ⇕ LCM( 5, 6)= 30 This matches our previous result. Ali and his grandma decide to plant six sunflowers per row.
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Diego and his mother are waiting for his father to arrive at a subway station. Diego notices that three subway lines arrived at the same time. The following table shows the arrival schedule of these three lines.
| Subway Line | Arrival Time |
|---|---|
| A | Every 12 minutes |
| B | Every 15 minutes |
| C | Every 10 minutes |
How long will Diego have to wait until all three lines arrive at the station together again?
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We are told that the three subways line arrived at the station at the same time. We want to know how long Diego will have to wait until the three lines meet at the station again. Let's look at the arrival times.
| Subway Line | Arrival Time |
|---|---|
| A | Every 12 minutes |
| B | Every 15 minutes |
| C | Every 10 minutes |
The next arrival of line A will be in 12 minutes. The train after that will arrive 12 minutes after that, and so on. The arrivals of line A are given by multiples of 12. Subway A: 12, 24, 36, 48, 60, 72, 84, ... We can apply this thought process to the other lines. Let's list the first multiples of each arrival time.
| Arrival Time | Multiples | |
|---|---|---|
| Line A | Every 12 minutes | 12, 24, 36, 48, 60, 72, 84, ... |
| Line B | Every 15 minutes | 15, 30, 45, 60, 75, 90, 105, ... |
| Line C | Every 10 minutes | 10, 20, 30, 40, 50, 60, 70, ... |
We can see from the table that the least common multiple of the three arrival times is 60. This means that Diego would have to wait for 60 minutes for the three lines arrive at the station together again. Wow! But before that, his father arrives and they can go home to have a great time together!
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Greatest Common Factor and Least Common Multiple
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