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Here are a few recommended readings before getting started with this lesson.
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.
Prime Factor  Product 

$2$  $80=40⋅2$ 
$2$  $40=20⋅2$ 
$2$  $20=10⋅2$ 
$2$  $10=5⋅2$ 
$5$  $5=5⋅1$ 
Prime Factorization  $80=2⋅2⋅2⋅2⋅5$ 
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 $19$ only once in each row and column and in each of the nine $3×3$ boxes.
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.
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.
greatest common divisorbecause a factor of a number divides that number evenly.
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.
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.
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?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.
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.
Each of these steps will now be applied.
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.
What a great achievement! Magdalena has made great progress on her puzzle.
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.
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:Multiples of3: 2,4,6,8,10,12,…3,6,9,12,15,… $

$6 ,12,18,24,…$  $LCM(2,3)=6$ 
$8$ and $12$  $Multiples of8:Multiples of12: 8,16,24,32,40,48,…12,24,36,48,… $

$24 ,48,72,96,…$  $LCM(8,12)=24$ 
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.
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 $540.$
Magdalena is having a great time solving her special Sudoku puzzle but the next clue looks scary.
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.
Begin by determining the prime factorization of each number. Label each factor in the prime factorizations that are repeated most. Multiply the labeled factors.
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.
Next, each of these steps will be performed.
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.
Lastly, bring down the factors of $10$ and $16$ in each column of the table created previously.
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 $80.$
Find the greatest common factor (GCF) or the least common multiple (LCM) of the given numbers, as requested.
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.
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.
She has $49$ roses and $84$ tulips. Each bouquet will have the same number of flowers and contain only roses or only tulips.
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.
Solve the following situations to help Magdalena complete that challenging puzzle.
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? 
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? 
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 — $4.$
Number  Multiples 

$6$  $6,$ $12,$ $18,$ $24,$ $30,$ $36,$ $42,$ $…$ 
$10$  $10,$ $20,$ $30,$ $40,$ $50,$ $60,$ $70,$ $…$ 
$5$  $5,$ $10,$ $15,$ $20,$ $25,$ $30,$ $35,$ $…$ 
Great news — the remaining numbers are easy to decipher and Magdalena has now filled in the rest of the board. What an outstanding achievement!