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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.
Number | Smallest Prime Factor | Quotient |
---|---|---|
80 | 2 | 280=40 |
40 | 2 | 240=20 |
20 | 2 | 220=10 |
10 | 2 | 210=5 |
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 1-9 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 of 2:Multiples of 3: 2,4,6,8,10,12,… 3,6,9,12,15,…
|
6,12,18,24,… | LCM(2,3)=6 |
8 and 12 | Multiples of 8:Multiples of 12: 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.