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Focus on CommonFactors and Multiples

Britannica Mathematics 1

Greatest Common Factor (GCF)A common factor is a whole number that divides evenly into two or more nonzerowhole numbers. The greatest common factor is the number with the greatest value ofall the factors. For example, the greatest common factor of 12 and 8 is 4, because 4 hasthe greatest value of all the numbers that divide into 12 and 8 evenly.

There are two common ways to find the greatest common factor of two numbers—listing the factors and using a factor tree.

Option 1: Listing the Factors

1. List the factors for each number.

2. Circle the factors that the numbers have in common.

3. Find the number with the greatest value in the list of common factors.

Consider the following problem: Find the greatest common factor (GCF) of 24 and 36.

First, list the factors for each number.

24: 1, 2 , 3 , 4 , 6 , 8 , 12, 24

36: 1, 2 , 3 , 4 , 6 , 9 , 12, 18, 36

Then, circle the factors that the numbers have in common.

24: 1, 2, 3, 4, 6, 8, 12, 24

36: 1, 2, 3, 4, 6, 9, 12, 18, 36

The common factors of 24 and 36 are 1, 2, 3, 4, 6, and 12.

Finally, find the common factor with the greatest value.

For 24 and 36, the greatest common factor is 12.

Focus on CommonFactors and Multiples

2 Common Factors and Multiples Student Material

Option 2: Using a Factor Tree

1. Find the prime factorization of each number using a factor tree. Notice that some ofthe numbers on the tree can be “broken down” into two smaller factors. Continuethe process until each factor is prime.

2. Circle the prime factors that the two numbers have in common.

3. Multiply the common prime factors to find the greatest common factor (GCF).

For example, for the numbers 24 and 36 start with the following trees.

Then circle the prime factors that 24 and 36 have in common.

The numbers 24 and 36 have the prime factors 3, 2, and 2 in common.

Finally, multiply the common prime factors to find the greatestcommon factor (GCF).

3 x 2 x 2 = 12

For 24 and 36, the greatest common factor is 12.

Focus on Common Factors and Multiples

24

6

3 2 2 2

4

36

6

3 2 3 2

6

24

6

3 2 2 2

4

36

6

3 2 3 2

6



Practice Finding the Greatest Common Factor Find the greatest common factor of each pair of numbers.

1. 6, 9

3. 18, 45

5. 15, 25

7. 36, 48

9. 17, 51

2. 12, 16

4. 24, 32

6. 18, 36

8. 24, 60

10. 60, 72


Solve each problem. Use scratch paper if needed to check your work.

11. Frank is running for class president. He is planning to give out 24 flyers and 20 buttons to promote himself. He wants each classroom to receive the same number of flyers and buttons, without having anything left over. What is the greatest number of classrooms to which Frank can hand out flyers and buttons?

13. Maria is making bouquets of flowers. She has 8 lilies and 16 daisies. Maria wants to make all the bouquets of flowers the same, without any flowers remaining. What is the greatest number of bouquets that Maria can make?

12. Melinda is making bags of supplies for the half-marathon that she is running with her friends. She has 16 water bottles and 20 apples. If the contents of each bag of supplies need to be exactly the same without having any bottles or apples left over, what is the greatest number of supply bags Melinda can make?

14. A school club has 14 boys and 21 girls. The teacher who sponsors the club wants to divide the club into groups, with each group having the same number of girls and boys. If all 35 of the club members need to be included in a group, what is the greatest number of groups the teacher can make?



Least Common Multiple (LCM)A multiple of a whole number is the product of the number and any nonzero wholenumber. A multiple that is shared by two or more numbers is a common multiple. Theleast common multiple of two or more numbers is the number with the least value. Forexample, the least common multiple of 4 and 6 is 12, because 12 is the number with theleast value when you list the multiples of both 4 and 6.

There are two common ways to find the least common multiple of two numbers—listing the multiples and using a factor tree.

Option 1: Listing the Multiples

1. List the first 10 multiples of each number.

2. Circle the multiples that the numbers have in common. If the two numbers do not have a multiple in common in their first 10 multiples, continue listing multiples untila common multiple is found.

3. Find the number with the least value in the list of common multiples.

Consider the following problem: Find the least common multiple of 6 and 8.

Start by listing the first 10 multiples for each number.

6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80

Then circle the multiples that the numbers have in common.

6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60

8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80

In their first 10 multiples, the common multiples of 6 and 8 are 24 and 48.

Finally, find the number with the least value.

For 6 and 8, the least common multiple is 24.



Option 2: Using a Factor Tree

1. Find the prime factorization of each number using a factor tree. Notice that eachnumber on the tree is “broken down” into two factors. Continue the process untileach factor is prime.

2. Write each prime factorization using exponents.

3. Write the prime factorizations as a single multiplication number sentence. If a number appears more than once, cross out the instance with the least exponent.

4. Multiply the remaining factors to find the least common multiple.

For example, for the numbers 6 and 8 start with the following trees:

Then, write each prime factorization using exponents:

6 = 3 x 2

8 = 23

Next, write the prime factorizations as a single multiplication sentence.

The number 6 can be written as the product of 3 and 2.

Cross out the repeated number with the least exponent:

3 x 2 x 23

Finally, multiply the remaining factors:

3 x 23 = 3 x 8 = 24

For 6 and 8, the least common multiple is 24.

6

3 2

8

4

2 2

2



Practice Finding the Least Common Multiple ProblemsFind the least common multiple of each pair of numbers.

1. 2, 3

3. 2, 5

5. 4, 7

7. 6, 9

9. 8, 12

2. 3, 4

4. 3, 6

6. 7, 8

8. 8, 10

10. 10, 12




11. An office supply store sells thank-you notes in packages of 9 and envelopes in packages of 12. If Jose wants to buy the same number of each, how many packages of envelopes will he need to buy?

13. Greg is thinking of a number that is evenly divisible by both 8 and 12. What is the least possible number that Greg can be thinking of?

12. Jason runs 7 miles every 4 days and swims 3 miles every 6 days. Today, Jason ran and swam. How many days from now will Jason go running and swimming on the same day again?

14. A bakery sells rolls in packages of 8 and bagels by the half-dozen. One day, the bakery sold exactly the same number of rolls and bagels. What is the least number of rolls and bagels the bakery could have sold?



The Distributive PropertyThe distributive property is a mathematical rule that states you can multiply a sum (ordifference) by multiplying each addend separately and then adding (or subtracting) theproducts.

For example, you can find the product of the following expression using thedistributive property.

3(6 + 5)

First distribute, or multiply, the 3 by each of the addends. Then add the newaddends together.

The sum (or difference) of two numbers can be rewritten as the product of a sum (ordifference) using a common factor. The distributive property can be used to checkyour work.

For example, you can factor the addends of the following sum and rewrite it as anequivalent expression.

12 + 20

1. List the factors for each number.

12: 1, 2, 3, 4, 6, 12

20: 1, 2, 4, 5, 10, 20

2. Circle the factors that the numbers have in common and find the greatest commonfactor.

12: 1, 2, 3, 4, 6, 12

20: 1, 2, 4, 5, 10, 20

The common factors of 12 and 20 are 1, 2, and 4. Their greatest common factor is 4.

3. Write the sum of 12 and 20 factoring out from each addend the greatest commonfactor of 4.

12 + 20

(4 x 3) + (4 x 5)

4(3 + 5)

The sum of 12 and 20 can be rewritten as 4(3 + 5). Notice that when checking yourwork by using the distributive property, each expression has the same value.

3 (6 + 5)

= 3 x 6 + 3 x 5

= 18 + 15

= 33



Practice Using the Distributive Property ProblemsRewrite each sum using the distributive property.

1. 10 + 15

3. 22 + 33

5. 20 + 48

7. 28 + 42

9. 72 + 88

2. 6(2 + 3)

4. 5(2 + 3)

6. 5(5 + 7)

8. 12(3 + 7)

10. 6(15 + 16)




11. A classroom has 7 tables. Each table has 4 chairs and 2 stools. How many places are there for the students to sit? Use the distributive property to find your answer.

13. Maureen is going on a picnic with her friends. She has 30 sandwiches and 45 pieces of fruit to share. Maureen wants to bring the food in picnic baskets, and she wants the contents of each picnic basket to be the same. Use the distributive property to find the greatest number of picnic baskets that Maureen would need to use so that each picnic basket has the same contents.

12. A pet store has 48 guppies and 36 goldfish. The manager at the pet store needs to put the guppies and goldfish in fish tanks, but he would like the same combination of guppies and goldfish in each tank. Use the distributive property to find the greatest number of fish tanks that the store manager would need to use so that each has the same combination of guppies and goldfish.

14. Seven friends are out riding their skateboards. Each has 1 helmet and 4 pads. How much safety equipment are the skateboarders wearing in all? Use the distributive property to find your answer.



VocabularyMatch the words to the clues.

1. factor

2. greatest common factor

3. prime number

4. prime factorization

5. product

6. multiple

7. least common multiple

8. distributive property

9. exponent

10. base

a. a number with the greatest value that divides into a set of numbers evenly

b. the product of a number and any nonzero whole number

c. the result of a multiplication problem

d. a whole number that divides evenly into a nonzero whole number

e. the mathematical property that states that ab + ac = a(b + c), where a, b, and c are real numbers

f. a number represented as the product of its prime factors

g. a number with exactly two factors, itself and 1

h. a multiple of two or more numbers with the least value

i. a number that is used as a factor in a repeated multiplication sentence

j. a number that shows how many times a base is used in a repeated multiplication sentence



factor

greatestcommon factor

prime number

primefactorization

product

multiple

least commonmultiple

distributiveproperty

exponent

base

Word or Phrase Definition or Description Picture or Example

Vocabulary


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