39.2: Common Multiples
Lesson
Let's use multiples to solve problems.
Exercise \(\PageIndex{1}\): 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?
Exercise \(\PageIndex{2}\): 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.
- 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.
- 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.
Exercise \(\PageIndex{3}\): Least Common Multiple
The least common multiple of 6 and 8 is 24.
- What do you think the term “least common multiple” means?
- Find all of the multiples of 10 and 8 that are less than 100. Find the least common multiple of 10 and 8.
- 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?
- What is the least common multiple of 10 and 20?
- What is the least common multiple of 4 and 24?
- 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.
Exercise \(\PageIndex{4}\): 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.
- 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.
-
Jada is the 36th customer.
- Will she get a prize? If so, what prize will she get?
- Is it possible for her to get more than one prize? How do you know? Explain your reasoning.
- How many prizes total will Lin’s uncle give away? Explain your reasoning.
Summary
A multiple of a whole number is a product of that number with another whole number. For example, 20 is a multiple of 4 because \(20=5\cdot 4\).
A common multiple for two whole numbers is a number that is a multiple of both numbers. For example, 20 is a multiple of 2 and a multiple of 5, so 20 is a common multiple of 2 and 5.
The least common multiple (sometimes written as LCM) of two whole numbers is the smallest multiple they have in common. For example, 30 is the least common multiple of 6 and 10.
One way to find the least common multiple of two numbers is to list multiples of each in order until we find the smallest multiple they have in common. Let's find the least common multiple for 4 and 10. First, we list some multiples of each number.
- Multiples of 4: 4, 8, 12, 16, 20 , 24, 28, 32, 36, 40 , 44 . . .
- Multiples of 10: 10, 20 , 30, 40 , 50, . . .
20 and 40 are both common multiples of 4 and 10 (as are 60, 80, . . . ), but 20 is the smallest number that is on both lists, so 20 is the least common multiple.
Glossary Entries
Definition: Common Factor
A common factor of two numbers is a number that divides evenly into both numbers. For example, 5 is a common factor of 15 and 20, because \(15\div 5=3\) and \(20\div 5=4\). Both of the quotients, 3 and 4, are whole numbers.
- The factors of 15 are 1, 3, 5 , and 15.
- The factors of 20 are 1, 2, 4, 5 , 10, and 20.
Definition: 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 \(3\cdot 10=30\) and \(5\cdot 6=30\). 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 . . .
Definition: Greatest Common Factor
The greatest common factor of two numbers is the largest number that divides evenly into both numbers. Sometimes we call this the GCF. For example, 15 is the greatest common factor of 45 and 60.
- The factors of 45 are 1, 3, 5, 9, 15 , and 45.
- The factors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15 , 20, 30, and 60.
Definition: 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 . . .
Practice
Exercise \(\PageIndex{5}\)
- A green light blinks every 4 seconds and a yellow light blinks every 5 seconds. When will both lights blink at the same time?
- A red light blinks every 12 seconds and a blue light blinks every 9 seconds. When will both lights blink at the same time?
- Explain how to determine when 2 lights blink together.
Exercise \(\PageIndex{6}\)
- List all multiples of 10 up to 100.
- List all multiples of 15 up to 100.
- What is the least common multiple of 10 and 15?
Exercise \(\PageIndex{7}\)
Cups are sold in packages of 8. Napkins are sold in packages of 12.
- 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?
- How many sets of cups and napkins will there be?
Exercise \(\PageIndex{8}\)
- Plot and connect these points to form a polygon.
\((-5,3),\qquad (3,3),\qquad (1,-2),\qquad (-3,-2)\)
- Find the lengths of the two horizontal sides of the polygon.
(From Unit 7.3.5)
Exercise \(\PageIndex{9}\)
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?
(From Unit 7.3.4)
Exercise \(\PageIndex{10}\)
A school wants to raise $2,500 to support its music program.
- If it has met 20% of its goal so far, how much money has it raised?
- If it raises 175% of its goal, how much money will the music program receive? Show your reasoning.
(From Unit 3.4.5)