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2. Greatest Common Factor and Least Common Multiple
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Chapter 1
2. 

Greatest Common Factor and Least Common Multiple

Student Learning Objectives:
  • Break a number into its prime factors
  • Identify the greatest common factor and least common multiple between two or more numbers
  • Understand and use the Distributive Property to simplify expressions

Why
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.

Problem Solving Reasoning and Communication Error Analysis Modeling Using Tools Precision Pattern Recognition
Lesson Settings & Tools
16 Theory slides
16 Exercises - Grade E - A
Each lesson is meant to take 1-2 classroom sessions
Image Credits expand_more
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. Although it sounds challenging, the greatest common factor and least common multiple are properties that are helpful 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.

Explore

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.

Rectangular arrangements using the factors of 12 and 18
Now, think about 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 in rectangles with the same width?
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 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
Any whole number greater than 1 can be factored into primes, and its factorization is unique. It should be noted that the prime factorization of a prime number is the number itself. Factor trees are also used to find prime factorizations.
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.

Factor Tree

Factor trees help express the prime factorization of a number by using the prime factors at the end of the branches.
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.

Sudoku Board

In this special version of the game, each cell contains a particular math challenge whose solution helps find the corresponding missing number. Magdalena is now focusing on 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 is the missing value for the red cell?

b The yellow cell must contain the most repeated factor of the prime factorization of 64. What is the missing value for the yellow cell?

Hint
a Make a factor tree of 54.
b Find the prime factorization of 64 by using a factor tree.

Solution
a To find the missing value on the red cell, begin by finding the prime factorization of 54. This can done with a factor tree by following these steps.
  1. Create a root node that contains the number 54.
  2. From the root node, draw two branches that a factor pair of 54.
  3. Break each factor of 54 into its own factor pair.
  4. Continue this process until only there are only prime factors on each branch.

With this information in mind, the factor tree of 54 can now be created.

Factor Tree of 54.

Now, consider that the missing value on the red cell is the most repeated prime factor of the prime factorization of 54. Prime Factorization of54 54=2* 3* 3* 3 Here, the number 3 is repeated three times, while the number 2 appears only once. This means that the missing value on the red cell is 3.

b Follow a similar procedure to find the prime factorization of 64.

Factor Tree of 64.

In this case, the prime factorization of 64 contains only the number 2 six times. Prime Factorization of64 64=2*2*2*2*2*2 Therefore, the missing value on the yellow cell of the Sudoku game is 2. Magdalena can now fill the yellow and red cells in the Sudoku board!

Red and Yellow Cells Filled
Discussion

Greatest Common Factor

Factors are used to divide a set of items into equal amounts. However, it may be difficult when two or more different sets are to be divided into a certain number of groups with equal amounts of each item. The common factors of the sets, 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.

Prime factorization of 4: 2 * 2. Prime factorization of 8: 2 * 2 * 2. The common prime factors (2 and 2) are shown within a circle. The product of these common prime factors is 4 (2 * 2).
It should be noted that the greatest common factor is also called the greatest common divisor because a factor of a number divides that number evenly.
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. Next, the common prime factors of the prime factorizations are identified. The GCF is then given by 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. In this case, the factor tree of 42 must be found.

Factor tree of 42.

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. Consider the factor tree of 24.

Factor tree of 24.

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 is factored as 2 * 3 * 7, and 24 is factored as 2 * 2 * 2 * 3. The common factors circled are one 2 and one 3.

4
Multiply the Prime Common Factors
expand_more
The greatest common factor of the numbers is given by 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.
Example

Find the GCF to Fill in a Set of Sudoku Cells

Magdalena continues with the process of solving this special version of the Sudoku game. This time, she wants to know which number goes in the purple cells.

Sudoku Board

The clue is that this purple cells must be filled with the greatest common factor (GCF) of 35 and 84. What number goes into the purple cells?

Hint
Find the prime factorization of 35. Next, find the prime factorization of 84. Identify the common prime factor between the two prime factorizations. Multiply the common prime factors to get the GCF.

Solution
The clue must be solved to help Magdalena figure out the number that goes in the purple cells. The clue claims that the number in the cells is given by the greatest common factor of 35 and 85. To find the GCF, these steps can be followed.

  1. Find the prime factorization of 35.
  2. Find the prime factorization of 84.
  3. Identify the common prime factors between the factorizations of 35 and 84.
  4. Multiply the common prime factors.

Each of these steps will now be applied.

Prime Factorization of 35

The prime factorization of 35 can be found by using a factor tree. The root node of the tree is 35 and then factor pairs of 35 will extend from the root node.

Factor Tree of 35

The prime factorization of 35 is 5* 7.

Prime Factorization of 84

A similar process can be followed to find the prime factorization of 84.

Factor Tree of 84

The prime factorization of 84 is 2* 2* 3* 7.

Identify Common Prime Factors

Now that the two prime factorizations are stated, find the common prime factor shared by these factorizations. &35=5* 7 &84=2*2*3* 7 The only common prime factors shared by these factorization is the number 7.

Multiply

In this case, only one factor is shared by the prime factorizations of 35 and 84, so the greatest common factor of 35 and 84 is 7. This number goes into the purple cells.

Purple cells filled in the Sudoku Board

What a great achievement! Magdalena has made great progress on her puzzle.
Explore

Comparing Multiples of 12 and 18

Consider the following applet that creates different arrangements of blocks using multiples of 12 and 18.

An applet showing different multiples of 12 and 18.
Now, consider the following questions and think about the possible answers to each.

  • Is there any multiples in common?
  • What is the least of the common multiples?
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. Some examples can be seen in the table below.

Numbers Multiples of Numbers Common Multiples Least Common Multiple
2 and 3 Multiples of2:& 2, 4, 6, 8, 10, 12, ... Multiples of3:& 3, 6, 9, 12, 15, ... 6, 12, 18, 24, ... LCM(2,3)= 6
8 and 12 Multiples of 8:& 8, 16, 24, 32, 40, 48, ... Multiples of12:& 12, 24, 36, 48, ... 24, 48, 72, 96, ... LCM(8,12)= 24
A special procedure exists for finding the LCM(a,b) of a pair of numeric expressions.
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 in the prime factorization that is repeated most. Finally, the LCM is given by the product of the highlighted factors. This process will be illustrated with this pair of numbers. 54 and 60 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.

Factor tree of 54.

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.

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.

Prime factorizations of 54 and 60 in a table.

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.

Factors of the prime factorizations of 54 and 60 in the table bringing down.

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.

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.

The Sudoku board with some orange and pink cells

The clue claims that to find the numbers to fill in the orange and pink cells, first 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.

Hint
Begin by determining the prime factorization of each number. Label each factor in the prime factorizations that are repeated most. Multiply the labeled factors.

Solution
Before finding the numbers to fill in the blue and pink cells, Magdalena first needs to find the least common multiple (LCM) of 10 and 16. Consider these steps to find the LCM.

  1. Determine the prime factorization of 10.
  2. Determine the prime factorization of 16.
  3. Write the prime factorization in a table to match prime factors vertically.
  4. Bring down the primes in each column and multiply them to get the LCM.

Next, each of these steps will be performed.

Prime Factorization of 10 and 16

Both prime factorizations will be found using factor trees. One tree will have 10 as its root, while the root of the second tree will be 16. Two branches extend from each root node to connect a factor pair. The process is repeated with the factors until only prime factors are on each branch.

Factor Tree of 10 and 16

Math Prime Factors Vertically

Now that the prime factorizations of 10 and 16 are found write them in a table. In the first row, place the prime factorization of 10 and the factorization of 16 in the second row. The table's columns will contain each factor while matching them vertically when possible.

A table displaying the prime factorization of 10 and 16, with common factors in the same column.

Bring Down the Primes in Each Column and Multiply

Lastly, bring down the factors of 10 and 16 in each column of the table created previously.

Prime Factors of 10 and 16 bring down in the table

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

The least common multiple of 10 and 16 is 80.

Finding the Numbers to Fill in the Orange and Pink Cells

Magdalena found that the LCM(10,16) is 80 but she is not ready to fill in the cells in the puzzle yet. For the orange cells, she needs to find the number that gives 80 when multiplied by 10. orange* 10=80 ⇒ orange= 8 The number that goes in the pink cells can be found in a similar way. pink*16=80 ⇒ pink= 5 This means that 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!

Purple cells filled in the Sudoku Board
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.

Applet generator of random number. It asks for the LCM or the GCF of the two numbers
Discussion

Distributive Property

An important use of the greatest common factor is that it can help simplify numeric expressions. Consider, for example, 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 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.

Distributive Property

The factor outside the parentheses is multiplied by, or distributed to, every term inside the parentheses. It should be noted that pulling out a common factor can make calculations more straightforward in some cases, but distributing that factor may be better in other calculations. Which way is better for the calculations will be dictated by the situation being studied.
Example

Creating Different Bouquets With the Same Amount of Flowers

Exhausted from solving math problems to solve the puzzle, Magdalena goes to the kitchen to have something to eat. She chooses a delicious slice of Italian pizza.

Slice of pizza

She is just about to return to her puzzle when her mother asks her for some help. 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.

A bouquet

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 and her mother can use in each bouquet?

b How many bouquets can Magdalena and her mother create?

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 and her mother want each of the bouquets to have an equal number of flowers. This number must be the greatest possible number of flowers. 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, find the prime factorizations of these numbers using a factor tree.

Prime Factorization of 49 and 84

Now that the prime factorizations have been found, look for the common prime factors between them. &49=7* 7 &84 = 2*2*3* 7 The only common factor between the factorizations is 7, so the GCF(49,84) is 7. This means that each bouquet will contain exactly 7 flowers!

b Begin by finding the total number of flowers.

49 roses + 84 tulips= 133 flowers Let c be the greatest possible number of flowers per bouquet. Next, let a be the number of rose bouquets and b the number of tulip bouquets. Note that the sum of a and b represents the total number of bouquets. Total Number of Bouquets: a+ b Additionally, multiplying the total number of bouquets by the number of flowers per bouquet c will equal the number of flowers in total. It was previously determined that there are 7 flowers per bouquet. Total Number of Flowers: c( a+ b) ⇓ 7( a+ b) The Distributive Property can be applied to this expression. 7( a+ b) ⇕ 7 a+ 7 b Now, consider that the number of rose bouquets a times seven will equal 49, the total number of roses. The missing value of a can be found using this information. Number of Rose Bouquets: 7* a=49 ⇓ a= 7 The number of tulip bouquets b can be determined by following a similar procedure. Number of Tulip Bouquets: 7* b=84 ⇓ b= 12 This means that Magdalena and her mother can create 7 rose bouquets and 12 tulip bouquets. Total Number of Bouquets: 7+ 12=19 bouquets

Checking Our Answer

 Checking if the Number of Bouquets for Each Flower Is Correct
Substitute the values of a and b into the original expression and simplify to verify Magdalena's calculations. Total Number of Flowers: c( a+ b) ⇓ 7( 7+ 12) This expression must equal the total number of flowers, 133. Use the Distributive Property to simplify this expression.

7( 7+ 12)
Distr

Distribute 7

7* 7+ 7* 12
Multiply

Multiply

49+84
AddTerms

Add terms

133

Notice that this is equivalent to multiplying the total number of bouquets, 7+12=19, by the total number of flowers per bouquet, 7. 19* 7= 133

Closure

Completing the Puzzle

After helping her mom and having a snack, Magdalena returns to her Sudoku puzzle. The board is almost done, but she thinks the next riddles look extremely difficult.

Sudoku Board

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?

b The green cells must be filled in with the solution to this riddle.

Davontay, Vincenzo, and Tadeo saw each other at the cinema today. If Davontay goes to the cinema every 6^(th) day, Vincenzo every 10^(th) day, and Tadeo every 5^(th) day, how many times must Tadeo go before the three friends will meet at the cinema again?

What is the number to fill in the green cells?

Hint
a List the factors of 100, 88, and 76. Use the list to find the GCF(100,88,76).
b List the first few multiples of 6, 10, and 5. Use the list to find the LCM(6,10,5). Find the number that, when multiplied by 5, equals the LCM(6,10,5).

Solution
a The greatest possible number of groups needs to be determined. The groups must meet these conditions.
  1. Each group must contain the same number of juniors.
  2. Each group must contain the same number of sophomores.
  3. Each group must contain the same number of seniors.

There are 100 juniors, 88 sophomores, and 76 seniors in the club. The coach wants to divide each grade level evenly to create the greatest number of groups possible. In other words, he wants to find the greatest common factor of these numbers. First, 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 the common factors 1, 2, and 4. Their greatest common factor is then 4. This means that the greatest number of groups that the coach can make is 4. This is also the number for the blue cells of Magdalena's puzzle — 4.

b The riddle says that Davontay goes to the cinema every sixth day. It will take 6 days for him to come to the cinema after today, 12 days from now for the second, and so on. This means that Davontay's visits are multiples of 6.

Davontay's Visits Multiples of6: 6, 12, 18, 24, 30, ... The same thought can be applied to Vincenzo's and Tadeo's situations. This means that the least common multiple of 6, 10, and 5 will give the number of days until the boys visit the cinema on the same day again. List the multiples of each to find the LCM(6,10,5).

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. This means it will take 30 days for the boys to visit the cinema on the same day. Now, recall that Tadeo goes to the cinema every five days. This means that 5 times a certain number must equal 30. 5* ?= 30 ? = 6 This number is 6. This means that Tadeo will go to the cinema 6 times before meeting his friends. This is the solution to the riddle, so Magdalena can fill in the green cells with the number 6.

Sudoku Board

Great news — the remaining numbers are easy to decipher and Magdalena has now filled in the rest of the board. What an outstanding achievement!

Sudoku Board solved



Level 1
Level 2
Level 3
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.

Rectangular box

If the dimensions of the box are whole numbers greater than 1, what are their possible dimensions?

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!

E
Hint & Answer
S
Solution

Dominika is creating identical fruit baskets using 56 bananas, 40 apples, and 65 oranges.

Bananas, apples, and oranges.

What is the greatest number of fruit arrangements she can make with the smallest amount of leftover fruit?

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.


E
Hint & Answer
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Solution

Consider the following statement.

The product of two different prime numbers is their least common multiple.

Is this statement always, sometimes, or never true?

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. ✓

E
Hint & Answer
S
Solution

Greatest Common Factor and Least Common Multiple

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