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.
| Student Learning Objectives: |
|---|
|
| | 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 in the red cell, we first need to find the prime factorization of 54. Let's make a factor tree to help us!
The factor that appears most often in the prime factorization of 54 is 3, so 3 goes into the red cell.
b
Let's find the prime factorization of 64 in the same way we did in Part A, using a factor tree.
Since the prime factorization of 64 contains only 2, it is the most repeated factor and goes into the yellow cell.
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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 Use a factor tree to determine the prime factorization of the first number.
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
To find the GCF of 35 and 84, let's first find the prime factorization of each number.
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 factors 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.
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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.
| 54 | 2 | 3 | 3 | 3 | ||
|---|---|---|---|---|---|---|
| 60 | 2 | 2 | 3 | 5 |
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.
| 54 | 2 | 3 | 3 | 3 | ||
|---|---|---|---|---|---|---|
| 60 | 2 | 2 | 3 | 5 | ||
| Factors | 2 | 2 | 3 | 3 | 3 | 5 |
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?
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Hint
Find the prime factorization of each number. Highlight each factor that is repeated most. Multiply the highlighted factors.
Solution
First, we need to find the least common multiple (LCM) of 10 and 16. Let's start by finding the prime factorization of each number using factor trees.
Now we can write the prime factors of 10 and 16 in a table and bring down the common factors. The product of these factors is the least common multiple.
| 10 | 2 | 5 | |||
|---|---|---|---|---|---|
| 16 | 2 | 2 | 2 | 2 | |
| Factors | 2 | 2 | 2 | 2 | 5 |
| LCM | 2*2*2*2*5 = 80 | ||||
The LCM of 10 and 16 is 80. For the orange cells, we need to find the number that gives 80 when multiplied by 10. For the pink cells, we need to find the number that gives 80 when multiplied by 16. lcr orange* 10=80 & ⇒ & orange= 8 pink*16=80 & ⇒ & pink= 5 We found that Magdalena needs to fill the orange cells with 8 and the pink cells with 5.
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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 84.
b
Write an expression for the total number of flowers.
Solution
a
Magdalena's mother wants each bouquet to have the greatest possible equal number of flowers, with only one type of flower in each bouquet. In other words, we need to find the greatest common factor of 49 and 84.
GCF(49,84)=? First, let's find the prime factorizations of these numbers using factor trees.
Now, let's look for the common factors between the prime factorizations. &49=7* 7 &84 = 2*2*3* 7 The only common factor in both factorizations is 7, so GCF(49,84) = 7. 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 each bouquet can have 7 flowers, so we can find the number of bouquets by dividing 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, with only one type of flower in each bouquet. Let's check our answer.
First, we find the number of rose bouquets by dividing the number of roses, 49, by the number of flowers per bouquet, 7.
49/7 = 7
Next, we find the number of tulip bouquets in the same way.
84/7 = 12
Magdalena's mother can make 7 rose bouquets and 12 tulip bouquets, so she can make 7 + 12 = 19 bouquets in total. This means our answer was 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 |
These numbers have 1, 2, and 4 as common factors, so 4 is the GCF. The greatest number of groups the coach can make is 4, so 4 goes in the blue cells.
b
The riddle says that Davontay goes to the movies every 6th day, Vincenzo every 10th day, and Tadeo every 5th day. This means their visits fall on multiples of 6, 10, and 5, respectively. 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 3.1
The volume V of a rectangular box is given by the product of its length l, width w, and height h. V=l* w* h A box has an volume of 1025 cubic centimeters.
If the dimensions of the box are whole numbers greater than 1, what are their possible dimensions?
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The volume of a rectangular box with length l, width w, and height h is given by the following formula. V=l* w* h We are told that a box has a volume of 1025 cubic centimeters. Let's substitute this information into the formula for the volume of a rectangular box. V&=l* w* h &⇓ 1025&=l* w* h We need to find 3 factors whose product is 1025, so let's find the prime factorization of 1025. This information can help us to know what the factors might be.
We have 5 factors to choose from: 1, 5, 5, 41, and 1025. We need to see what combinations of 3 of these factors can make 1025.
| Combinations | Simplification |
|---|---|
| 1*1*1025= 1025 | 1*1*1025= 1025 |
| 5*5*41= 1025 | 5*5*41= 1025 |
| 1*(5*5)*41= 1025 | 1*25*41= 1025 |
| 1*5*(5*41)= 1025 | 1*5*205= 1025 |
We found all the combinations of these factors that make 1025. However, recall that all the dimensions of the box must all be greater than 1. Let's reconsider our table with this in mind.
| Combinations | Valid Dimensions? |
|---|---|
| 1* 1*1025=1025 | * |
| 5*5*41=1025 | ✓ |
| 1*25*41=1025 | * |
| 1*5*205=1025 | * |
We can then say that the possible dimensions of the box are 41, 5, and 5 centimeters.
Notice that it does not matter which dimension any of the values is assigned to — the box can be 41 centimeters in height, length, or width, so long as the other two dimensions are 5 centimeters!
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Dominika is creating identical fruit baskets using 56 bananas, 40 apples, and 65 oranges.
What is the greatest number of fruit arrangements she can make with the smallest amount of leftover fruit?
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We are asked to find the greatest number of identical fruit baskets that Dominika can make with the least amount of fruit left over. Consider the amount of fruit she has. &Bananas: &56 &Apples: &40 &Oranges:& 65 The greatest common factor of these numbers is the number of identical baskets that Dominika can make without having any leftover fruit. Let's list the factors of each of these numbers to find the common factors and choose the greatest one.
| Factors | |
|---|---|
| 56 | 1, 2, 4, 7, 8, 14, 28, 56 |
| 40 | 1, 2, 4, 5, 8, 10, 20, 40 |
| 65 | 1, 5, 13, 65 |
The only common factor between these numbers is 1, but it does not really make sense for Dominika to make only one arrangement with all the fruit in it. Since there are no other factors common to all three three numbers, any other number of arrangements will end up with some fruit left over. Let's identify any factors common to at least two of the numbers to minimize the amount of fruit leftover.
| Factors | |
|---|---|
| 56 | 1, 2, 4, 7, 8, 14, 28, 56 |
| 40 | 1, 2, 4, 5, 8, 10, 20, 40 |
| 65 | 1, 5, 13, 65 |
Dominika could make 5 and 8 arrangements. If she makes five arrangements, there will be some bananas left over. If she makes eight baskets, there will be some oranges left over. Let's find out how much fruit Dominika has on hand at the start of the basket production process. Total Fruit: 56 + 40 + 65 = 161 Now let's see which option produces the least amount of leftover fruit.
| Fruit Per Basket | Total Fruit Used | Fruit Left Over | |
|---|---|---|---|
| 5 Baskets | 11 bananas, 8 apples, 13 oranges | 11* 5+8* 5+13* 5= 160 | 161- 160=1 |
| 8 Baskets | 7 bananas, 5 apples, 8 oranges | 7* 8+5* 8+11* 8= 160 | 161- 160=1 |
Notice that both options have 1 piece of fruit left over. However, since 8 is greater than 5, the most amount of baskets that Dominika can make with the least amount of fruit leftover is 8 baskets.
We can use the applet to determine the amount of fruit for each basket option.
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Consider the following statement.
|
The product of two different prime numbers is their least common multiple. |
Is this statement always, sometimes, or never true?
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We are asked if the product of two different prime numbers gives the least common multiple (LCM) of these numbers. Let's begin by finding the LCM of some pairs of prime numbers. This will give us an idea about the truthfulness of the given statement. Let's consider the factor trees of 13 and 17.
Notice that the prime factors of each of these numbers are the numbers themselves and that these factors appear only once. This means that the product of these numbers is their least common multiple. LCM(13,17)&= 13* 17 &⇕ LCM(13,17)&=221 Now let's look at the prime factors of 47 and 97.
The situation for this pair of numbers is the same — the only prime factors are the numbers themselves. Let's multiply them to find their LCM. LCM(47,97)&= 47* 97 &⇕ LCM(47,97)&=4559 Since the only factors of a prime number are 1 and the number itself, the only prime factor of a prime number is that number itself. This means that the product of two different prime numbers gives us their LCM. This means that the statement is always true.
The product of two different prime numbers is their least common multiple. ✓
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Greatest Common Factor and Least Common Multiple
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