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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 and

The following applet creates arrangements using the factors of and 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 and 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 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.
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
Number Smallest Prime Factor Quotient
Prime Factorization
Any whole number greater than 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 grid with some cells containing numbers and others blank. The purpose of the game is to find the missing values by using the numbers only once in each row and column and in each of the nine 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 What is the missing value for the red cell?
b The yellow cell must contain the most repeated factor of the prime factorization of What is the missing value for the yellow cell?

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

Solution
a To find the missing value on the red cell, begin by finding the prime factorization of This can done with a factor tree by following these steps.
  1. Create a root node that contains the number
  2. From the root node, draw two branches that a factor pair of
  3. Break each factor of 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 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
Here, the number is repeated three times, while the number appears only once. This means that the missing value on the red cell is
b Follow a similar procedure to find the prime factorization of
Factor Tree of 64.
In this case, the prime factorization of contains only the number six times.
Therefore, the missing value on the yellow cell of the Sudoku game is 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 and
The common factors of and are and The 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.
The 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 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
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 and 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 and are and
Therefore, the greatest common factor of and is
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 and What number goes into the purple cells?

Hint

Find the prime factorization of Next, find the prime factorization of 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 and To find the GCF, these steps can be followed.

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

Each of these steps will now be applied.

Prime Factorization of

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

Prime Factorization of

A similar process can be followed to find the prime factorization of
Factor Tree of 84
The prime factorization of is

Identify Common Prime Factors

Now that the two prime factorizations are stated, find the common prime factor shared by these factorizations.
The only common prime factors shared by these factorization is the number

Multiply

In this case, only one factor is shared by the prime factorizations of and so the greatest common factor of and is 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 and

Consider the following applet that creates different arrangements of blocks using multiples of and
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 and is the smallest whole number that is a multiple of both and It is denoted as The least common multiple of and is the smallest whole number that is divisible by both and Some examples can be seen in the table below.

Numbers Multiples of Numbers Common Multiples Least Common Multiple
and
and
A special procedure exists for finding the 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.
The 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 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
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 and 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 and 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
and

The least common multiple of and which can also be written as is

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 and Next, these conditions must be met.

  • The number for the orange cells multiplied by is the
  • The number for the pink cells multiplied by is the
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 and Consider these steps to find the LCM.

  1. Determine the prime factorization of
  2. Determine the prime factorization of
  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 and

Both prime factorizations will be found using factor trees. One tree will have as its root, while the root of the second tree will be 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 and are found write them in a table. In the first row, place the prime factorization of and the factorization of 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 and 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 and

Factors Multiplication
and

The least common multiple of and is

Finding the Numbers to Fill in the Orange and Pink Cells

Magdalena found that the is 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 when multiplied by
The number that goes in the pink cells can be found in a similar way.
This means that the number that goes in the orange cells is and the number that goes in the pink cells is 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.
The greatest common factor of these numbers is This means that each number can be rewritten using this common factor.
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.
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 roses and 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 and
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 and
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.
The only common factor between the factorizations is so the is This means that each bouquet will contain exactly flowers!
b Begin by finding the total number of flowers.
Let be the greatest possible number of flowers per bouquet. Next, let be the number of rose bouquets and the number of tulip bouquets. Note that the sum of and represents the total number of bouquets.
Additionally, multiplying the total number of bouquets by the number of flowers per bouquet will equal the number of flowers in total. It was previously determined that there are flowers per bouquet.
The Distributive Property can be applied to this expression.
Now, consider that the number of rose bouquets times seven will equal the total number of roses. The missing value of can be found using this information.
The number of tulip bouquets can be determined by following a similar procedure.
This means that Magdalena and her mother can create rose bouquets and tulip bouquets.

Checking Our Answer

 Checking if the Number of Bouquets for Each Flower Is Correct
Substitute the values of and into the original expression and simplify to verify Magdalena's calculations.
This expression must equal the total number of flowers, Use the Distributive Property to simplify this expression.
Distr

Distribute

Multiply

Multiply

AddTerms

Add terms

Notice that this is equivalent to multiplying the total number of bouquets, by the total number of flowers per bouquet,
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 juniors, sophomores, and 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 day, Vincenzo every day, and Tadeo every 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 and Use the list to find the
b List the first few multiples of and Use the list to find the Find the number that, when multiplied by equals the

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 juniors, sophomores, and 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

Notice that these numbers have the common factors and Their greatest common factor is then This means that the greatest number of groups that the coach can make is This is also the number for the blue cells of Magdalena's puzzle —

b The riddle says that Davontay goes to the cinema every sixth day. It will take days for him to come to the cinema after today, days from now for the second, and so on. This means that Davontay's visits are multiples of
The same thought can be applied to Vincenzo's and Tadeo's situations. This means that the least common multiple of and 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
Number Multiples
The least common multiple of and is This means it will take 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 times a certain number must equal
This number is This means that Tadeo will go to the cinema times before meeting his friends. This is the solution to the riddle, so Magdalena can fill in the green cells with the number
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



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