Answer Key: Unit 1 Special Divisibility Rulessupportcdn.edmentum.com/InstructorMaterials/DevMath1PreAlg/dl... · Divisibility rules help to determine if one number is divisible by

Copyright © 2007 PLATO Learning, Inc. All rights reserved. PLATO® is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning are trademarks of PLATO Learning, Inc. PLATO, Inc. is a PLATO Learning, Inc. company.

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Pre-Algebra A

Unit 1

Answer Key: Special Divisibility Rules

Name _________________________________________ Date____________________________

Objective In this activity, you will use the divisibility rules for 2, 3, 4, 5, 6, and 10. Materials

• Pencil or pen • Highlighter

Warm-Up On the hundreds chart to the right, do the following:

• Highlight the numbers that are divisible by 2. Circle the numbers that are divisible by 3.

• Cross out the numbers that are divisible by 6.

(The first two rows have been done for you.) Do you notice any patterns?

On the hundreds chart to the right, do the following:

• Highlight the numbers that are divisible by 5. • Circle the numbers that are divisible by 10.

Do you notice any patterns?

Study Using any method you choose, determine which of these numbers evenly divide the whole numbers below. List each number that evenly divides the whole numbers.

2, 3, 4, 5, 6, 10

Example: 232 ______________________

Copyright © 2007 PLATO Learning, Inc. All rights reserved. PLATO® is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning are trademarks of PLATO Learning, Inc. PLATO, Inc. is a PLATO Learning, Inc. company. 2

232 ÷ 2 = 116 Yes 232 ÷ 3 = 77.333 No 232 ÷ 4 = 58 Yes 232 ÷ 5 = 46.4 No 232 ÷ 6 = 38.66 No 232 ÷ 10 = 23.2 No

155 ______________________________ 5, 10 2, 4

966 ______________________________

520 ______________________________ 2, 3, 6 2, 4, 5, 10

363 ______________________________ 344 ______________________________ 2, 4 3

Divisibility rules help to determine if one number is divisible by another number. You probably already know a few divisibility rules. For example, 44 is divisible by 2 because 44 is an even number, and all even numbers are divisible by 2. You will be using a few of the more common divisibility rules throughout the lesson. Use the list you made in the step above, along with what you know about dividing numbers, to decide which divisibility rule goes with which number below. Then draw a line to connect the divisibility rule with its matching number. Some numbers may go with more than one rule, but each rule should have a number, so find the best match.


Using your divisibility rule matches from above, circle all the numbers in the right column that evenly divide the numbers in the left column. Check your answers with a calculator. If necessary, change your divisibility rule matches from above. You may circle more than one number.


Use what you know about divisibility rules to write them in your own words. If you know any other divisibility rules, add them to the bottom of the page.

Divisibility rule for 2:

If a number is even, then I know it can be divided by 2.


If I add up all the digits in a number and that number is divisible by 3, then I know

the first number is divisible by 3.


If the number ends in a 0 or a 5, then I know it is divisible by 5.


If a number is divisible by both 2 and 3, then it is also divisible by 6.


If the number ends in a 0, then I know it is divisible by 10.

Divisibility rule for _____:

Answers will vary depending on the number stated above.

Divisibility rule for _____:

Answers will vary depending on the number stated above.


Wrap-Up Fill in the blanks with the missing words or numbers to complete the divisibility rules. Example: Divisibility rule for 2:

Any whole number that has a _____, _____, _____, , or _____ in the ones place can be evenly divided by 2.

0 2 4 6 8


If the ________ of the digits for a whole number is divisible by 3, then the whole number is also divisible by 3.

sum


If the number formed by the last ______ digits of the whole number is divisible by 4, then the whole number is also divisible by 4.

two



Any whole number that has a ____ or a ____ in the ones place can be evenly divided by 5. 0 5


If a number satisfies the divisibility rules for both _____ and ______, then it is also divisible by 6.

2 3


Any whole number that has a _____ in the ones place can be evenly divided by 10. 0


18

60

50

15

45

30

20

75

Answer Key: Finding Factors and Prime

Factors

Pre-Algebra A Unit 2

Name _________________________________________ Date____________________________

Objective In this activity, you will find the factors and the prime factors of whole numbers. Materials Ruler Vocabulary factor: a whole number that can be multiplied by another whole number to form a product

(Example: 2 and 3 are factors of 6 because 2×3=6)

prime number: a whole number that has only 1 and the number itself as factors (Examples: 2, 5, 11)

composite number: a whole number that has factors other than 1 and the number itself (Example: 9)

=75) prime factorization: a list of prime factors that form a certain product (Example: 3×5×5 Warm-Up The numbers 2, 3, and 5 are prime numbers. You can’t divide them by any whole numbers except 1 and themselves. That means the multiplication statements below are the only possible ones with these prime numbers as the product. (Remember, 1 is not considered a prime or a composite number.)

1×2 2 or 2×1=2 1×3 3 or 3= = ×1=3 1×5 = 5 or 5×1=5 We multiplied these prime numbers by each other. The products are the composite numbers in the column on the right. Match the multiplication problem with the products by writing them in the blanks.

5×5×2 = ____ 5×5×3 = ____ 3×3×5 = ____

5×2×2×3 = ____ 2×5×3 = ____ 2×3×3 = ____

5×3 = ____ 2×5×2 = ____

18 60 50 15 45 30 20 75

You’ve just completed the prime factorization of these composite numbers.

Study Arrays and Factors These arrays help to show the factors of 15. You can arrange 15 rectangles into the arrays shown. Fill in the missing numbers that go with each array.

=15 1 15 ____×____

3 5 _____×_____ 15 =

1 3 5 15 Now use the arrays to list all the factors of 15: _____ _____ _____ _____

noWhat if you start with only 3 or with only 5 rectangles? Can you arrange 3 or 5 rectangles in other ways and have the same number of columns in each row? ______ 1×3 3 1= ×5=5

(student work space) Explain why you can or cannot rearrange the rectangles. The numbers 3 and 5 do not have other factors, so you can’t divide them into other arrays.

1, 3 1, 5 List the factors of 3 and 5. Factors of 3: _____________ Factors of 5: _____________ Copyright © 2007 PLATO Learning, Inc. All rights reserved. PLATO® is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning are trademarks of PLATO Learning, Inc. PLATO, Inc. is a PLATO Learning, Inc. company. 2

Now let’s look at the number 20. We’ll use arrays to find its factors. 1×20=20

Fill in the missing numbers for this array of 20 rectangles.

2 10 _____×_____ 20 (Accept factors in either order.) =

The number 20 has one more set of factors that isn’t shown in the arrays above. Draw an array in the space below to show the missing set of factors. Then fill in the missing numbers. (You may want to use a ruler to draw the array.)

4 5 ____×____ 20 (Accept factors in either order.) =

(Sample array. Accept any 4×5 or 5×4 array.)

1, 2, 4, 5, 10, 20 Now use the arrays to list all the factors of 20: ___________________________ Dividing to Find Factors You can also find the factors of a number without drawing arrays. Start with the smallest numbers and check if they are factors, and then work up to bigger numbers. Keep a list of the factors as you go. Let’s see how this works with 12. You know 1 is a factor, and 1×12=12. So you’d put 1 and 12 on the list. Check to see if 2 is a factor. It is because 2×6=12. So you add 2 and 6 to the list of factors. Check to see if 3 is a factor. It is because 3×4=12. So you add 3 and 4 to the list of factors. We already listed 4 as a factor, 5 is not a factor, and 6 is on the list already. If we check 7, 8, 9, 10, and 11 we’ll find that they aren’t factors either. We already have 12 on the list. So the factors of 12 are 1, 2, 3, 4, 6, and 12. Copyright © 2007 PLATO Learning, Inc. All rights reserved. PLATO® is a registered trademark of PLATO Learning, Inc. Straight Curve, Academic Systems, and PLATO Learning are trademarks of PLATO Learning, Inc. PLATO, Inc. is a PLATO Learning, Inc. company. 3

Use the same process to find the factors of 54. First list the smaller numbers 54 is divisible by and the factor that number goes with. (Use as many of the blanks as you need.) Then list all the factors in the blank below.

Factors of 54:

1 54 _____×_____=54

27 2 _____×_____=54

3 18 _____×_____=54

6 9 _____×_____=54 _____×_____=54

1, 2, 3, 6, 9, 18, 27, 54 _____×_____=54 All the factors of 54: ________________________________ Prime Factors Let’s review prime numbers before we find the prime factors of composite numbers. Here’s a reminder of what prime and composite mean.

Prime number: a number that has only two factors, which are 1 and the number itself. This means that it is not divisible by any whole numbers except 1 and itself.

Composite number: a number that has more than two factors. This means that it is divisible

by other numbers than 1 and itself. Underline all the prime numbers in this list: 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 27 28 29 Note about the number 1: The number 1 is not considered prime or composite. Because it has only one factor—itself—it’s neither. That means it won’t be included in a list of prime factors of a number. So, the factors of 6, for example, are 1, 2, 3, and 6. But the prime factors of 6 are just 2 and 3.


Factor Trees One way to find prime factors is to use a factor tree. To factor the number 80, for example, we start dividing it until we end up with only prime numbers. Here are two ways to make a factor tree for 80. 2 × 2 × 2 × 2 × 5 = 80 2 × 5 × 2 × 2 × 2 = 80 In the factor tree on the left, we divided 80 by the prime number 2 until we got the prime number 5. In the factor tree on the right, we divided by composite factors of 80 and factored down until each number was the product of prime numbers. We circled each prime number so that it is easier to list the prime factors from the factor tree. Finish these factor trees for 126 by filling in the missing numbers. Then write the prime factorization for 126 in the blank below.

2 × 3 × 3 × 7_____________________________________=126 (Accept factors in any order.)


Which factor tree did you find more useful for finding the prime factors of 126? Explain. Answers will vary. Sample answer:

I think the factor tree on the left is more useful. I like to divide the number by primes first. Then I

can see all the primes as I divide the numbers in each part of the tree.

Dividing This is another way to find the prime factors of a number. Instead of using a tree to show the division steps, you can divide a number repeatedly. Start with small prime numbers and see if they are factors of the number. Let’s find the prime factors of 84. We’ll start with familiar small prime numbers like 2, 3, and 5.

yes yes noDo you think 84 is divisible by 2? ____ Do you think it’s divisible by 3? ____ By 5? ____ Let’s start by dividing 84 by prime numbers so we can list it as a product of primes. Find solutions for the division problems below. Rewrite each solution in the first blank of the next line to continue factoring 84.

42 84 2 ____ ÷

42 21

=

÷ =

÷ =

(That result) ____ 2 ____

21 7 (That result) ____ 3 ____ Your last result should be a prime number. It is also a factor of 84. Now, rewrite 84 as the product of each of the numbers you divided it by and your last result.

2 2 7 3 84 ____×____×____×____ This is the prime factorization of 84. =

Note: There is no correct order to list prime factors. But it is conventional to list the smallest primes factors first and then the higher ones (be sure to include every factor, even if it occurs more than once). So the prime factorization of 18 should be shown as 2×3×3 18 rather than 3×2×3 18.

==


Practice 1. Make a factor tree for the numbers below. Then write the prime factorization of each number in

the blank.

(Sample factor tree shown)

=24 (Accept factors in any order.) 2 x 2 x 2 x 3______________________________


2 x 2 x 3 x 3______________________________=36 (Accept factors in any order.)


2 x 5 x 7______________________________=70 (Accept factors in any order.)


2 x 23______________________________=46 (Accept factors in any order.)


2. Which list shows only prime numbers? 2. Which list shows only prime numbers? a) 2 5 7 11 13a) 2 5 7 11 13 b) 2 5 7 9 11 c) 3 4 7 8 11 d) 2 5 6 7 11

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

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

1, 2, 5, 7, 10, 14, 35, 70

1, 2, 23, 46

3. List all the factors of these numbers. Factors of 24: __________________________________ Factors of 36: __________________________________ Factors of 70: __________________________________ Factors of 46: __________________________________


Wrap-Up Wrap-Up Take the numbers 100 and 32. Draw a factor tree to write the prime factorization for one of the numbers, and use the division method to find the prime factorization for the other number. Take the numbers 100 and 32. Draw a factor tree to write the prime factorization for one of the numbers, and use the division method to find the prime factorization for the other number. Prime factorization of 100: (Method used for 100: __________________) Prime factorization of 100: (Method used for 100: __________________)

(Method and display of factor tree or division will vary.) 2 2 5 5 100 (Accept factors in any order.) × × × =

Prime factorization of 32: (Method used for 32: ___________________)

(Method and display of factor tree or division will vary.)

2 2 2 2 2 32 (Accept factors in any order.) × × × × = Now list all the factors for both numbers.

1, 2, 4, 5, 10, 20, 25, 50, 100 Factors of 100: _________________________ (Accept factors in any order.) 1, 2, 4, 8, 16, 32 Factors of 32: _________________________ (Accept factors in any order.)

Finally, describe how prime factorization can help you find all the factors of a number. Answers will vary. Sample answer:

Prime factorization helps because you can combine the prime factors to make the other factors.

But you can still miss some factors. For example, you have to combine 2×25 50 to see that 50 is =

a factor of 100 from the prime factor list. So I like to match each small factor with a large factor to

see if I missed any. For 100, for example, I checked that I had 2 and 50 because 2 50× =100.


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Answer Key: Unit 1 Special Divisibility Rulessupportcdn.edmentum.com/InstructorMaterials/DevMath1PreAlg/dl... · Divisibility rules help to determine if one number is divisible by