__Objective __

At the end of this lesson, students should be able to:

- Interpret and solve word problems involving HCF.
- Interpret and solve word problems involving LCM.

__Example 1__

The Preston Corporation would like to donate 18 computers and 12 printers to local schools. The corporation would like to make sure that each school receives the same set of computers and printers with none left over. What is the greatest number of schools that The Preston Corporation can donate to?

__Solution__

The greatest common factor is the greatest whole number that is a factor of each of two or more numbers.

List the factors of each number. Find the largest number that appears in both lists.

Factors of 18:1, 2, 3,** 6**,9,18

Factors of 12:1, 2, 3, 4, **6**, 12

The greatest common factor of 18 and 12 is **6**. That means that the greatest possible number of schools is 6, because the 18 computers could be given to 6 schools with each school receiving 3 computers and the 12 printers could be given to 6 schools with each school receiving 2 printers.

The greatest number of schools The Preston Corporation can donate to is **6**.

__Example 2__

Brandy is packing crates that can hold 4 pears, and Emilia is packing crates that hold 12 pears. If they must pack the same number of pears, what is the minimum number of pears each must pack?

__Solution__

The least common multiple is the least whole number that is a multiple of each of two or more numbers.

You need to find the smallest number that is a multiple of both** 4** and **12**. This is the least common multiple.

Write the prime factorization for each number.

**4** = **2 **× **2**

**12** = **2 **× **2 **× **3**

Repeat each prime factor the greatest number of times it appears in any of the prime factorizations above.

**2 **× **2** × **3** = **12**

The least common multiple of** 4** and **12** is **12****.** That means that the minimum number of pears each must pack is **12**, because 3 crates of 4 pears packed by Brandy is 12 pears in total and 1 crate of 12 pears packed by Emilia is 12 pears in total.

The smallest number of pears that each must pack is 12.

__Example 3__

The yearbook editor wants each page of the Activities section of the yearbook to have the same combination of color photos and black-and-white photos. If there are 15 color photos and 10 black-and-white photos, all of which the editor wants to use, what is the greatest number of Activity pages the editor can create?

__Solution __

The greatest common factor is the greatest whole number that is a factor of each of two or more numbers.

Write the prime factorization for each number.

15 = 3 × 5

10 = 2 × 5

Next, find the common factors shared by both of the numbers.

15 = 3 × **5**

10 = 2 × **5**

The only common factor of 15 and 10 is 5, so the greatest common factor is **5**. That means that the greatest possible number of pages is 5, because 15 color photos could be put onto 5 pages with 3 color photos each and 10 black-and-white photos could be put onto 5 pages with 2 black-and-white photos each.

The greatest number of Activities pages the editor can create is 5.

__Example 4__

By coincidence, Kimberly’s Paint Supplies sold equal quantities of two paint colors yesterday: yellow and blue. All yellow paint comes in 9-liter containers while blue paint is sold in 5 liter containers. What is the smallest amount of each paint color the store must have sold.

__Solution__

The least common multiple is the least whole number that is a multiple of each of the two numbers.

You need to find the smallest number that is a multiple of both **9** and** 5**. This is the least common multiple.

Write the prime factorization for each number. 5 is a prime number.

**9** = **3** ×** 3**

**5 ****= ****5 **×** 1**

Repeat each prime factor the greatest number of times it appears in any of the prime factorizations above.

**3** × **3** × **5 **=** 45**

The least common multiple of **9 **and **5** is **45**. That means that the smallest amount of each color is 45 liters, because 5 9-liter containers of yellow paint is a total of 45 liters of yellow paint and 9 5-liter containers of blue paint is a total of 45 liters of blue paint.

The smallest amount of each paint color is 45 liters.

Lesson Content

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