# Illustrative Mathematics Unit 6.7, Lesson 17: Common Multiples

Learning Targets:

• I can explain what a common multiple is.
• I can explain what the least common multiple is.
• I can find the least common multiple of two whole numbers.

Related Pages
Illustrative Math

#### Lesson 17: Common Multiples

Let’s use multiples to solve problems.

Illustrative Math Unit 6.7, Lesson 17 (printable worksheets)

#### Lesson 17 Summary

The following diagram explains what is a common multiple and how to find the least common multiple of two whole numbers. #### Lesson 17.1 Notice and Wonder: Multiples

Circle all the multiples of 4 in this list.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
Circle all the multiples of 6 in this list.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26
What do you notice? What do you wonder?

Scroll down the page for the solutions to the “Are you ready for more?” section.

#### Lesson 17.2 The Florist’s Order

A florist can order roses in bunches of 12 and lilies in bunches of 8. Last month she ordered the same number of roses and lilies.

1. If she ordered no more than 100 of each kind of flower, how many bunches of each could she have ordered? Find all the possible combinations.
2. What is the smallest number of bunches of roses that she could have ordered? What about the smallest number of bunches of lilies? Explain your reasoning.

#### Lesson 17.3 Least Common Multiple

The least common multiple of 6 and 8 is 24.

1. What do you think the term “least common multiple” means?
2. Find all of the multiples of 10 and 8 that are less than 100. Find the least common multiple of 10 and 8.
3. Find all of the multiples of 7 and 9 that are less than 100. Find the least common multiple of 7 and 9.

#### Are you ready for more?

1. What is the least common multiple of 10 and 20?
2. What is the least common multiple of 4 and 12?
3. In the previous two questions, one number is a multiple of the other. What do you notice about their least common multiple? Do you think this will always happen when one number is a multiple of the other? Explain your reasoning.

If one number is a multiple of the other then the least common multiple is the larger number.
If we list out the multiples of the numbers, we will find that the larger number is the least common multiple because it is a multiple of the smaller number.

#### Lesson 17.4 Prizes on Grand Opening Day

Lin’s uncle is opening a bakery. On the bakery’s grand opening day, he plans to give away prizes to the first 50 customers that enter the shop. Every fifth customer will get a free bagel. Every ninth customer will get a free blueberry muffin. Every 12th customer will get a free slice of carrot cake.

1. Diego is waiting in line and is the 23rd customer. He thinks that he should get farther back in line in order to get a prize. Is he right? If so, how far back should he go to get at least one prize? Explain your reasoning.
2. Jada is the 36th customer.
a. Will she get a prize? If so, what prize will she get?
b. Is it possible for her to get more than one prize? How do you know? Explain your reasoning.
3. How many prizes total will Lin’s uncle give away? Explain your reasoning.

#### Glossary Terms

common multiple
A common multiple of two numbers is a product you can get by multiplying each of the two numbers by some whole number. For example, 30 is a common multiple of 3 and 5, because 3 · 10 = 30 and 5 · 6 = 30. Both of the factors, 10 and 6, are whole numbers.
The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33 . . .
The multiples of 5 are 5, 10, 15, 20, 25, 30, 35, 40 . . .
The common multiples of 3 and 5 are 15, 30, 45, 60 . . .

least common multiple
The least common multiple of two numbers is the smallest product you can get by multiplying each of the two numbers by some whole number. Sometimes we call this the LCM. For example, 30 is the least common multiple of 6 and 10.
The multiples of 6 are 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 . . .
The multiples of 10 are 10, 20, 30, 40, 50, 60, 70, 80 . . .

#### Lesson 17 Practice Problems

1. a. A green light blinks every 4 seconds and a yellow light blinks every 5 seconds. When will both lights blink at the same time?
b. A red light blinks every 12 seconds and a blue light blinks every 9 seconds. When will both lights blink at the same time?
c. Explain how to determine when 2 lights blink together.
2. a. List all multiples of 10 up to 100.
b. List all multiples of 15 up to 100.
c. What is the least common multiple of 10 and 15?
3. Cups are sold in packages of 8. Napkins are sold in packages of 12.
a. What is the fewest number of packages of cups and the fewest number of packages of napkins that can be purchased so there will be the same number of cups as napkins?
b. How many sets of cups and napkins will there be?
4. Rectangle ABCD is drawn on a coordinate plane. A = (-6,9) and B = (5,9).
What could be the locations of points C and D?
5. A school wants to raise \$2,500 to support its music program.
a. If it has met 20% of its goal so far, how much money has it raised?
b. If it raises 175% of its goal, how much money will the music program receive? Show your reasoning.

The Open Up Resources math curriculum is free to download from the Open Up Resources website and is also available from Illustrative Mathematics.

Try the free Mathway calculator and problem solver below to practice various math topics. Try the given examples, or type in your own problem and check your answer with the step-by-step explanations. 