divisibility rules - bradburydunlop /...

Divisibility Rules Triclcs that help us determine if a number is a multiple of another number are called divisibility rules.

CIRCLE ALL THE CORRECT EXAMPLES

2 divides any number ending with an even digit (2,4,6,8,0). Examples) 32 1234 774 10106

3 divides any number when the sum ofthe digits is divisible by 3. Examples) 312 658 1130622 3232

4 divides any number where the last two digits form a number divisible by 4.

Examples) 924 132 10234 7482

5 divides any number ending in 5 or 0. Examples) 105 10575 90637 72

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6 divides any number divisible by 2 AND 3. Examples) 32 312 1130622 774

7 divides any number where the last digit, multiplied by 2 and subtracted from the remaining digits is divisible by 7.

Examples) 287 331 455 875

8 divides any number where the last three digits are divisible by 8. Examples) 5008 5387 3244 331008

9 divides any number where the sum ofthe digits is divisible by 9. Examples) 5994 381921 504 775

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Divisibility Rules

A number Is divisible by . . .

2

,3

4

,5

6

8

9

10

If...

the last digit is even (0, 2, 4, 6, or 8)

the sum of the digits is divisible by 3

the number formed hy the last two digits is divisible hy 2 at least twice

the last digit is 0 or 5

the number is divisible hy both 2 and .^

the number is divisible by 2 at least three times

the sum of the digits is divisihie hy 9

the last digit is 0

Numbers cannot be divided by 0.

You can use the divisibility rules to find factors of a number.

You can write fractions in lowest terms by dividing the numerator and the denominator by common factors until the only common factor is 1.

CMmniintaMte tht Mtai

1. a) Why is a number that is divisible by 6 also divisible by 2 and 3?

b) A number is divisible by 10. What other numbers is the number divisible by? How do you know?

2. a) Explain one method for determining the greatest common

factor of 36 and 20.

b) Share your answer with a partner.

3. Simone wrote - ^ in lowest terms as -j-r.

i) Is she finished yet? Explain. 18 b) Show a method for writing ~ in lowest terms.

4. Explain what you know about divisibility by 0. Include an example in your explanation.

f i 206 MHR. Chapter 6

1.2 More Patterns in Division

Quick Review >- A number is divisible by 3 if the sum of its digits is divisible by 3. For example,

1035 is divisible by 3 because 1 + 0 + 3 + 5 = 9, and 9 is divisible by 3. 1036 is not divisible by 3 because 1 + 0 + 3 + 6 = 1 0 , and 10 is not divisible by 3.

>- A number is divisible by 6 if the number is divisible by 2 and by 3. For example, 1038 is divisible by 2 because the number is even. 1038 is divisible by 3 because 1 + 0 + 3 + 8=12 , which is divisible by 3. So, 1038 is divisible by 6.

V A number is divisible by 9 if the sum of its digits is divisible by 9. For example, 5418 is divisible by 9 because 5 + 4 + 1 + 8 = 18, and 18 is divisible by 9. 5428 is /tot divisible by 9 because 5 + 4 + 2 + 8 = 19, and 19 is ttof divisible by 9.

>" No number is divisible by 0.

>- You can use a Carroll diagram to show numbers that are divisible by two numbers. This Carroll diagram shows divisibility by 6 and by 9.

Divisible by 9 Not divisible by 9

Divisible by 6 18,36, 126, 162 6, 12,204,402

Not divisible by 6 27,45,963,711 10, 29, 325, 802

>- You can use divisibility rules to help list the factors of a number. To list the factors of 156: Try each rule in turn. Divide by 2: 156 + 2 = 78 Divide by 3: 156 + 3 = 52 Divide by 4: 156 + 4 = 39 156 is not divisible by 5. Divide by 6: 156 + 6 = 26 156 is not divisible by 7, by 8, by 9, or by 10. Use a calculator to check for divisibility by 11 and 12. 156 is not divisible by 11. Divide by 12: 156 + 12 = 13

Since the factors 12 and 13 are close in value, you have found all the factors. In order, the factors of 156 are: 1, 2, 3, 4, 6, 12, 13, 26, 39, 52, 78, 156

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Practice

1. Circle the numbers that are divisible by 2.

23 98 21 44 11 77 34


55 10 7 59 105 775 1025

3. Circle the numbers that are divisible by 2 and by 5.

10 30 25 55 1000 52

4. Write each number in the correct place in the Venn diagram. 16, 20, 33, 64, 80, 95, 97, 105, 214, 216, 324, 405

5. Write four 3-digit numbers that are divisible by 10.

Divisible by 4 Divisible by 5

6. Write three 4-digit numbers that are divisible by 8.

7. a) Write each number in the correct place in the Venn diagram. 115, 116, 120, 168, 450, 753, 800,928,1008,1110

Divisible by 8

b) Write 4 more numbers in the Venn diagram - one in each loop and one outside the loops. How do you know you placed each number correctly?

Divisible by 10

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Practice

1. Match the niunber with the correct divisibility statement. Draw more than one line if it is needed.

54 Divisible by 10.

56 Divisible by 3.

50 Divisible by 9.

92 Divisible by 8.

75 Divisible by 5.

93 Divisible by 2.

30 Divisible by 6.

2. Cross out the numbers that are not divisible by 2.

12 79 98 134 227 2469

How do you know the numbers are not divisible by 2?


91 331 333 153 99 12 321

How do you know you are correct?

4. Write four numbers that are divisible by 6: How did you choose those numbers?

5. Solve each riddle.

a) I am divisible by 2 and by 3. I am between 21 and 29. Which number am I?

21 22 23 24 25 26 27 28 29

b) I am divisible by 5 and by 10. I am between 56 and 64. Which number am I?

c) I am divisible by 2 and by 9. I am between 424 and 449. Which number am I?

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6, Which numbers below are divisible by 3? By 6? By 9? How do you know?

a) 124 „________ _____„ _„

b) 215

c) 330

d) 450

e) 150

7. Use your answers to question 6 to help you list the factors of each number.

a) 124:

b) 215:

c) 150:

8- a) Sort these numbers in the Carroll diagram below. 16, 18,27,37, 120, 180,281,288,352,411,432,540

Divisible by 4

Not divisible by 4

Divisible by 9 Not divisible by 9

b) Write one more number in each part of the Carroll diagram. Explain how you knew where to place each number.

9. a) Sort these numbers in the Venn diagram. 12, 28, 36, 54, 72, 79, 135, 256, 270, 318, 371, 432

b) Which loop is empty? Explain why there is no number that belongs in that loop.

Multiples of 4 Multiples of 6

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reproduce this page is restricted to the purchasing school.

Multiples of 9

Practise

For help with #5 to #8, refer to Example 1 on page 202.

5. Which of the following numbers are

divisible by 5? Explain how you know.

1010 554 605 902 900 325

6. Which of the following numbers are

divisible by 4? Explain how you know.

124 330 3048 678 982 1432

7. a) Use a diagram or table to sort the numbers according to divisibility by 4 and 8.

312 330 148 164 264 13 824

b) If a number is divisible by 4 and 8, what is the smallest number other than I that it is also divisible by? How do you know?

8. a) Using a diagram or table, sort the numbers based on divisibility by 6 and 10.

5832 35 010 243 9810 3 1 9 9 0 b) If a number is divisible by 6 and 10,

what is the smallest number other than 1 that it is also divisible by? How do you know?

For help with #9 to #14, refer to Example 2 on page 203.

9. Use the divisibility rules to list the factors of the following numbers.

a) 36 b) 15 c) 28

10. What are the factors of these numbers?

a) 18 b) 54 c) 72

11. Use the divisibility rules to determine the common factors for each pair of numbers.

a) 3 and 6

b) 4 and 8

c) 6 and 12

12. What are the common factors for each pair of numbers?

a) 5 and 10

b) 4 and 12

c) 24 and 15

13. a) Use the divisibility rules to determine the common factors of 16 and 20. Include a Venn diagram as part of your answer.

b) What is the greatest common factor of 16 and 20?

14. a) What are the common factors of 10 and 30? Include a Venn diagram with your answer.

b) Identify the greatest common factor of 10 and 30.

For help with #15 and #16, refer to Example 3 on pages 204-205.

15. Write the following fractions in lowest terms.

a)

d)

ii 20 _9_ 12

b)

e)

6_ 18

_4_ 10

c)

f)

10 16 9_ 15

16. Write each fraction in lowest terms.

16 ' ' M 2 ' ' 20 14 .. 5 ,, 12

a)

d) 24

e) 0

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6.1 Divisibility. MHR 207

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1.8 Factors and Divisibility MATHPOWERTW Seven, pp. 26-27

The factors of a number each divide the number evenly.

4S - 1 = 48 48 - 2 = 24 48 + 3 = 16 48 - 4 = 12 48 - 6 = 8 48 ~ 48 = 1 48 ^ 24 = 2 48 - 16 = 3 48 - 12 = 4 48 -- 8 --- 6

The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The divisibility rules can help you find the factors of a number. A number is divisible by 2 if it ends in 0, 2, 4, 6, or 8. 6 if it is divisible by 2 and 3. 3 if the sum of the digits is divisible by 3. 8 if the last thr€>e digits are divisible by 8. 4 if the last two digits are divisible by 4. 9 if the sum of the digits is divisible by 9. 5 if it ends in 0 or 5. 10 if it ends in 0.

15. 65: 1, ,.__, 13, _^

16. SO: _ , 2, _, 5,.._, H), _ , 20, ___, 80

State the mi

L 5 X

3. 11 X

5. X

7. X

Write tivo p

9. X

10. X

11. X

12. X

13. X

Complete ea

14. 32: 1,2,

ssing factor.

_ = 4 5

:= 77

7 - 4 9

4 = 28

2.

4.

6.

8.

airs of factors for

= 2 4 ;

- 2 6 ;

= 4 8 ;

- 8 0 ;

- 108 ,

zh Ust of factors.

Q in

2 X

9 X

each

_ X _

X

X

X

,_ y--

X 3 = 21

X 6 = 54

- 16

- 3 6

number.

- 24

--- 26

^ 48

= 80

- 108

23. In questions 18-22, which numbc^rs

a) are divisible bv 2?

b) are divisible bv 3?

c) are divisible bv 2 and 3?

State the missing factors. Use factors greater than 1.

24. 2 X X - 180

25. X X =: 300

26. X 4 X = 240

27. 8 X X - 192

28. X X 2 = 168

29. X 9 X ^ 225

30. X X - 212

Find the smaUest number ivhose factors are

31. 3. 4. and 5.

•VJ 1 A 3nf{ <

—

17 . 3, 4, 9, 12, 18, __,

l:i'-t aU the fictors nf eacn number.

J 8. -2

19. '"-6

33. 11, 4, and 2.

34. 10, 2, and 6.

Use the divisibdity rules to determine which of the f}nlovzing numbers are drvisibk tn/ 8 nnd zvhich are divisible bu 9.

21.

35. 729 __

37. 14 112

36. 1520

38. Tnere are 24 desks in the classroom. In how manv wavs can the desks be arranged in equal groups?

i

Copyright X '19s<n N-fcCXaw-HiJl Rver-^cn Urnite-d

Name: Date:

Prime or Composite?

Classify each number ai Rox 1

1. 27 L-jPrime nComposite

5. 4 5 ^ •Pr ime t-JComposite

Box 2

1. 18 •p r ime nComposite

5. 55 L-IPrime i—IComposite

j2.

6.

2.

»

i prime or composite.

26 3. DPrime LJComposite

15 ]7 . •p r ime •composite

11 T3. DPrime | •composite j

2 |7. •p r ime •composite

37 •p r ime •composite




|4.

J8.

|4.

8.





Box 3 1. 12

•p r ime I nComposite

15. 42 •p r ime •composite

2.

6.

3 |3. •Prime I •composite I

48 ]7. •p r ime I •composite |





Box 4 il. 97

•p r ime •Composite


2. 67 •Prime •composite

~'6r u " ^ •p r ime •composite

13. 59 I •pr ime I •composite

V- 71 ^ j •p r ime 1 •composite



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Name: Date:

Place Value

Arranging numbers in place value charts can help with estimation. The place value chart below shows 1247.63.

Thousands Hundreds Tens Ones Decimal Point Tenths Thousandths

The number 1248.63 is one more than 1247.63. o o ^ C ^ k ^ J ^ ^ h e o n ^ ^

The number 1147.63 is one hundred less than 1247 63 •oo

The number 1247.83 is two tenths more than 1247.63. o

1. Arrange the numbers in the place value chart.

a) 1349.52

b) 45.069

c) 100.05

d) 0.455

Thousands Hundreds Tens Ones Oecimat

Point Tenths

—

Hundredths Thousandths

Compare and Order Numbers

You can use a place value chart to compare and order numbers. The numbers 270, 2.7, and 27 are shown in the following place value chart.

270 2.7

27

Hundreds 2 0 0

Tens 7 0 2

Ones 0 2 7

• • •

Tenths 0 7 0

The numbers arranged from greatest to least are 270 27, and 2.7. You can write this as 270 > 27 > 2.7

O O O O O O O

rhe symbol > means "Is greater than."

2. Arrange each set of numbers from greatest to least.

a) 1.8,2.8,1.9

3. Use the symbol > to show the numbers arranged from greatest to least.

b) 365.7,358, 365.9

14 MHR • Chapter 2: Operations on Decimal Numbers

a) 1.9, 2.4, 2 b) 5, 4 .3, 0.7

• ^

I)

N a m e

1.3 Place Value MATHPOWERTM Seven, pp . 8-9

A place value table can help you read and write numbers .

7 'S^l TS7 Read decim.al numbers by naming the ,'' Thousands .' Ones

' ^ / c- /• ^ / S/ c / eSf / / $•'/ S 7 # / A'-^^/

>c

8 j 7 i 6 t 2 L l 5 9 6 3

place value of the final digit.

0.247 -^ ti vo hundred fort\ '-seven thousand ths

Read " and" for the decimal point,

29 375.56 -^ twenty-nine thousand three hundred seventy-five and fifty-six hundred ths

State the total value ofeach imderlined digit.

1. 23.45

2. 19 789 456

3. 457.3864

Write each nurnber in standard form.

4- eight thousand three h u n d r e d two

5. two million thirty-three thousand five hundred four

6. six and five tenths

7. one thciusand thirteen and eight hundred fortv-nine thousandths

Write each number in ivords.

8. 2894

9. 687.95

10. 0.35:

11. 1976.089

Wnte each number in standard fomi.

12. 10 000 - 3000 + 8

13. 4 X 1 000 000 - 6 X in 000 X- 9 X 1000 6 X 100 -r 4 X 1

14. 3 X 10 + 5 X 1 + 4 X 0.1 + 2 X 0.01

15. 300 + 41,1 4- 2 f 0.1 + 0.08 + 0.007

Write in expanded form.

16. 203

17. 34.127

18. 276.13

19, 34 123 006

Insert > , =, or < to make each statement true

20. 34.56 n 30 -r 4 + 0.05 + 0.006

21. 12 309 • 10 000 + 2000 + 300 4- 90

22. 35.7 Q 30 X 0.05 - 0,007

23. 5.01 D 5 X 0.01

24. The points awarded to the winners m the men's Olympic platform diving competition are show^n. Order them from, highest to lowest.

Year

1972

Points

504.12

Year | Points

1984 T 710.91

1976

1980 600.51 1 88 ^^5.65 1 quo

638.61

677.31

CfspvritdU X 1996 McGraw-Hill Ryerson iJmiteci

Name

1.9 Problem Solving: Make an Assumption MATHPOWER™ Seven, pp. 30-31

Unders t and the Problem

Carry Out the Plan

Look Back

Determine a pattern and make an assumption. Then, list the next 3 terms.

1. 100, 98, 94, 88,

2. 15,21,27,33,

3. 0, 3, 7, 10, ^

4. The Kerels paid $98.84 for hydro m March this year. How much will they pay for hydro in one year? Wliat assumptions have you made?

5. Connor rode his bicycle 3 km in 4 min. How far could he ride in 2 h? WTiat assumptions have vou made?

6. The student council purchased 6 cases of cola, 2 cases of'ginger ale, and 2 ca.ses of orange soda for the dance. What assumptions did they make?

7. Each week, Mr. Blake purchases two 4-Lbags of milk for his family. This week, milk cost $3.29 for a bag. How much will he spend on milk in a year? WTiat a.ssumptions have vou made?

For eadi pair of numbers, name a third number that could be next in the sequence.

9. 9,, 6, __

11. 12, 24,

10. 3, 8,

12. 2, 8,

13. a) What assumptions did you make in questions 9-12 to determine each pattern?

b) Suggest a different third number for each original pair of numbers in questions 9-12. What assumptions have you made this time?

14. A citv councillor conducttHl a suney. She asked every 10th household whether they were in favour of the city developing a community playground in the neighbourhood. The results showed that 65'?<! were in favour of the plan.

a) What assumptions might the councillor make from. Iier surx'ev^

x fn her ji*b at the supermarket, [ei^nv :o unpack a case nf 48 cans of soup. She hc>ught she vvould stack them m 2 lnvin vith H row- ; of 3 cans in each iaver, Wha assumptions did she nia,ke?

lad b) Do you think the sun-'ev is accurate for the u'hoie neighibourhotxl? Explain.

Lovvn-sbl h^ McC.Xatva-iiJi Kyerson Lrmiiec

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