Ch 02

Multiples, factors and primes

2 A room measures 550 centimetres by 325 centimetres. What would be the side length of the largest square tile that can be used to tile the floor without any cutting? This chapter will provide you with skills to answer this question more easily.

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Maths Quest 7 for Victoria

Multiples A multiple of a number is the answer obtained when that number is multiplied by another whole number. All numbers in the 5 times table are multiples of 5, so 5, 10, 15, 20, 25, . . . are all multiples of 5. The number 5 has been multiplied by one other number to find each of the numbers in the 5 times table, so they are all multiples of 5.

WORKED Example 1 List the first 5 multiples of 7. THINK

WRITE

1

First multiple is the number × 1 = 7 × 1.

2

Second multiple is the number × 2 = 7 × 2.

3

Third multiple is the number × 3 = 7 × 3.

4

Fourth multiple is the number × 4 = 7 × 4.

5

Fifth multiple is the number × 5 = 7 × 5.

7, 14, 21, 28, 35

WORKED Example 2 Write the numbers in the list that are multiples of 8. 18, 8, 80, 100, 24, 60, 9, 40 THINK

WRITE

1

The biggest number in the list is 100. List multiples of 8 using the 8 times table just past 100; that is, 8 × 1 = 8, 8 × 2 = 16, 8 × 3 = 24, 8 × 4 = 32, 8 × 5 = 40, etc.

8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88, 96, 104

2

Write any multiples that appear in the list.

Numbers in the list that are multiples of 8 are 8, 24, 40, 80.

remember remember 1. A multiple of a number is the answer obtained when that number is multiplied by another whole number. For example, all numbers in the 5 times table are multiples of 5; that is, 5, 10, 15, 20, 25 . . . 2. If 2 or more numbers have the same multiple, it is called a common multiple.

Chapter 2 Multiples, factors and primes

2A WORKED

Example

1

WORKED

Example

2

43

Multiples

1 List the first 5 multiples of the following numbers. a 3 b 6 c 100 e 15 f 4 g 21 i 14 j 12 k 50 m 33 n 120 o 45

d h l p

2.1

11 25 30 72

2 Write the numbers in the following list that are multiples of 10. 10, 15, 20, 100, 38, 62, 70 3 Write the numbers in the following list that are multiples of 7. 17, 21, 7, 70, 47, 27, 35

Multiples

4 Write the numbers in the following list that are multiples of 16. 16, 8, 24, 64, 160, 42, 4, 32, 1, 2, 80 5 Write the numbers in the following list that are multiples of 35. 7, 70, 95, 35, 140, 5, 165, 105, 700 6 The numbers 16, 40 and 64 are all multiples of 8. Find 3 more multiples of 8 that are less than 100. 7 a List the first 10 multiples of 4. b List the first 10 multiples of 6. c In your lists, circle the multiples that 4 and 6 have in common (that is, circle the numbers that appear in both lists). d What is the lowest multiple that 4 and 6 have in common? This is the Lowest Common Multiple of 4 and 6, known as the LCM. 8 a b c d

Lowest common multiple

List the first 6 multiples of 3. List the first 6 multiples of 9. Circle the multiples that 3 and 9 have in common. What is the lowest common multiple of 3 and 9?

9 Find the LCM of each of the following pairs of numbers. a 3 and 6 b 5 and 7 c d 9 and 6 e 6 and 15 f g 7 and 10 h 9 and 12 i j 10 and 25 k 12 and 16 l

6 and 8 4 and 12 15 and 20 4 and 15

10 Answer true (T) or false (F) to each of the following statements. a The LCM of 4 and 5 is 20. b The LCM of 3 and 6 is 18. c 20 is a multiple of 10 and 2 only. d 15 and 36 are both multiples of 3. e 60 is a multiple of 2, 3, 6, 10 and 12. f 100 is a multiple of 2, 4, 5, 10, 12 and 25. g 30 is the LCM of 6 and 5.

Lowest common multiple

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Maths Quest 7 for Victoria

11 multiple choice a The first 3 multiples of 9 are: A 1, 3, 9 B 3, 6, 9 b The first 3 multiples of 15 are: A 15, 30, 45 B 1, 3, 5, 15 c The LCM of 6 and 9 is: A 6 B 54 d The LCM of 7 and 12 is: A 21 B 84

C 9, 18, 27

D 9, 18, 81

E 1, 9, 18

C 30, 45, 60

D 1, 15, 30

E 45

C 36

D 9

E 18

C 48

D 1

E 12

12 Place the first 6 multiples of 3 into the triangle at right, so each line adds up to 27. Use each number once only.

number once only.

13 Kate goes to the gym every second evening while Ian goes every third evening. If they both attended the gym on Monday, how long will it be before they both attend again on the same day? What day of the week will this be? 14 Vinod and Elena are riding around a mountain bike trail. Each person completes one lap in the time shown on the stopwatches.

MIN

SEC

100/SEC

05:00:00

MIN

SEC

100/SEC

07:00:00

If they both begin cycling from the starting point at the same time, how long will it be before they pass this starting point again at exactly the same time?

Chapter 2 Multiples, factors and primes

45

15 Two smugglers, Bill Bogus and Sally Seadog have set up signal lights that flash continuously across the ocean. Bill’s light flashes every 5 seconds and Sally’s light flashes every 4 seconds. If they both start together, how long will it take for both lights to flash again at the same time? 16 Alex and Nadia were having races running down a flight of stairs. Nadia took the stairs 2 at a time while Alex took the stairs 3 at a time. In each case, they reached the bottom with no steps left over. a How many steps are there in the flight of stairs? List 3 possible answers. b What is the smallest number of steps there could be? c If Alex can also take the stairs 5 at a time with no steps left over, what is the smallest number of steps in the flight of stairs?

Multiples, factors and primes 01

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GAME time

17 Twenty students in Year 7 were each given a different number from 1 to 20 and then asked to sit in numerical order in a circle. Three older girls, Milly, Molly and Mandy came to distribute jelly beans to the class. Milly gave red jelly beans to every 2nd student, Molly gave green jelly beans to every 3rd student and Mandy gave blue jelly beans to every 4th student. a Which student had jelly beans of all 3 colours? b How many students received exactly 2 jelly beans? c How many students did not receive any jelly beans?

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1 I am a 2-digit number that can be divided by 3 with no remainder. The sum of my digits is a multiple of 4 and 6. My first digit is double my second digit. What number am I? 2 Find a 2-digit number such that if you subtract 3 from it, the result is a multiple of 3; if you subtract 4 from it, the result is a multiple of 4 and if you subtract 5 from it, the result is a multiple of 5. 3 In a class election with 3 candidates, the winner beat the other 2 candidates by 3 and 6 votes respectively. If 27 votes were cast, how many votes did the winner receive?

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Maths Quest 7 for Victoria

In 1893, which country countr y was the first to let women women vote? vote? 24 44 27 48 10 60 40 14 65 28

Which people wer were e not allow allowed to vote until 1924? 18 36

6

8

12

15 100 9

34 30 45 20 22 42 21

Find the code to answer these questions by matching the letter beside each pair of numbers below with their lowest common multiple (LCM).

A

2, 7

D

5, 9

I

6, 4

N

8, 6

C 3, 15

D

7, 4

A

11, 2

N

3, 9

I

10, 4

L 8, 10

E

2, 5

N 10, 6

A 5, 12

M 4, 18

I

17, 2

N 6, 14

E

2, 3

R

4, 8

A

S

7, 3

W 3, 27

Z 16, 3

E

4, 11

N 13, 5

9, 2

A 4, 25

Chapter 2 Multiples, factors and primes

47

Factors A factor is a whole number that divides exactly into another whole number, with no remainder. If one number is divisible by another number, the second number divides exactly into the first number. For example, 8 is divisible by 4 because 8 ÷ 4 = 2. The number 4 is a factor of 8 because 4 divides into 8 twice with no remainder, or 8 ÷ 4 = 2. The number 2 is also a factor of 8 because 8 ÷ 2 = 4. The number 3 is a factor of 15 because 3 goes into 15, or 15 ÷ 3 = 5.

WORKED Example 3 Find all the factors of 14. THINK 1

2

3

WRITE

1 is a factor of every number and the number itself is a factor; that is, 1 × 14 = 14. 14 is an even number so 14 is divisible by 2 and is a factor. Divide the number by 2 to find the other factor (14 ÷ 2 = 7). Write a sentence placing the factors in order from smallest to largest.

1, 14

2, 7

The factors of 14 are 1, 2, 7 and 14.

Factor pairs It is often easiest to write factors in pairs called factor pairs. These are pairs of numbers which multiply to equal a certain number.

WORKED Example 4 List the factor pairs of 30. THINK 1 2 3

4 5

1 and the number itself are factors; that is, 1 × 30 = 30. 30 is an even number so 2 and 15 are factors; that is, 2 × 15 = 30. Divide the next smallest number into 30. Therefore, 3 and 10 are factors; that is, 3 × 10 = 30. 30 ends in 0 so 5 divides evenly into 30, that is, 5 × 6 = 30. List the factor pairs.

WRITE 1, 30 2, 15 3, 10

5, 6 The factor pairs of 30 are 1, 30; 2, 15; 3, 10 and 5, 6.

48

Maths Quest 7 for Victoria

Common factors A common factor is a number which is a factor of two or more given numbers. If the numbers 6 and 10 are given, then 2 is a common factor because 2 is a factor of 6 and 2 is a factor of 10. The Highest Common Factor or HCF is the largest of the common factors.

WORKED Example 5

a Find the common factors of 8 and 24 by: i listing the factors of 8 ii listing the factors of 24 iii listing the factors common to both 8 and 24. b State the highest common factor of 8 and 24. THINK

WRITE

a iii

a iii 1, 8 2, 4 Factors of 8 are 1, 2, 4, 8. iii 1, 24 2, 12 3, 8 4, 6 Factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24. iii Common factors are 1, 2, 4, 8.

Find the pairs of factors of 8. Write them in order. iii 1 Find the pairs of factors of 24. 2 Write them in order. iii Write the common factors. 1

2

b Find the highest common factor.

b HCF is 8.

Factors can make it easier to multiply numbers mentally. The following examples show the thought processes required.

WORKED Example 6

Find 12 × 15 using factors. THINK 1 2 3 4

WRITE

Write the question. Write one of the numbers as a factor pair. Multiply the first two numbers. Find the answer.

12 × 15 = 12 × 5 × 3 = 60 × 3 = 180

WORKED Example 7

Use factors to evaluate 32 × 25. THINK 1 2

3

WRITE

Write the question. Rewrite the question so that one pair of factors is easy to multiply. The number 4 is a factor of 32 and 4 × 25 = 100. Find the answer.

32 × 25 = 8 × 4 × 25 = 8 × 100 = 800

With practice this could be done in your head. Some useful products are 2 × 50 = 100, 8 × 125 = 1000, 4 × 250 = 1000.

Chapter 2 Multiples, factors and primes

49

remember remember 1. A factor is a whole number that divides exactly into another whole number, with no remainder. 2. The factors of a number can be found using factor pairs. One of the pairs may be a single number which multiplies by itself. The number 5 is a factor of 25 because 5 × 5 = 25. The number itself and 1 are always factors of a number. 3. If 2 or more numbers have the same factor, it is called a common factor. 4. The highest common factor or HCF of 2 or more numbers is the largest factor which divides into all of the given numbers.

2B WORKED

Example

3

WORKED

Example

4

Factors

1 Find all the factors of each of the following numbers. a 12 b 8 c 40 e 28 f 60 g 100 i 39 j 85 k 76 m 99 n 250 o 51

d h l p

2 List the factor pairs of: a 20 b 18

d 132

c

36

35 72 69 105

Factors

2.2

3 If 3 is a factor of 12, state the smallest number greater than 12 which has 3 as one of its factors. WORKED

Example

5

4 a Find the common factors of 15 and 35 by: i listing the factors of 15 ii listing the factors of 35 iii listing the factors common to both 15 and 35. b State the highest common factor of 15 and 35. 5 a List the factors of 21. b List the factors of 56. c Find the highest common factor of 21 and 56. 6 a List the factors of 27. b List the factors of 15. c Find the highest common factor of 27 and 15. 7 a List the factors of 7. b List the factors of 28. c Find the highest common factor of 7 and 28. 8 a List the factors of 48. b List the factors of 30. c Find the HCF of 48 and 30. 9 Find the highest common factor of 9 and 36. 10 Find the highest common factor of 26 and 65. 11 Find the highest common factor of 42 and 77.

Highest common factor

Highest common factor

50

Maths Quest 7 for Victoria

12 Find the highest common factor of 24 and 56. 13 Find the highest common factor of 36 and 64. 14 Find the highest common factor of 18 and 72. 15 Find the highest common factor of 45, 72 and 108. 16 multiple choice a A factor pair of 24 is: A 2, 4 B 4, 6 b A factor pair of 42 is: A 6, 7 B 20, 2 c The HCF of 12 and 30 is: A 2 B 3 d The HCF of 15 and 33 is: A 1 B 15

C 6, 2

D 2, 8

E 1, 14

C 21, 1

D 16, 3

E 0, 42

C 30

D 6

E 12

C 3

D 5

E 33

17 Which of the numbers 3, 4, 5 and 11 are factors of 2004? 18 Find the following using factors. a 12 × 25 b 12 × 35 6 d 11 × 16 e 11 × 14 g 20 × 15 h 20 × 18

c f i

12 × 55 11 × 15 30 × 21

19 Use factors to evaluate the following. a 36 × 25 b 44 × 25 7 d 72 × 25 e 124 × 25 g 56 × 50 h 48 × 125

c f i

24 × 25 132 × 25 52 × 250

WORKED

Example

WORKED

Example

20 Connie Pythagoras is trying to organise her Year 7 class into rows for their class photograph. If Ms Pythagoras wishes to organise the 20 students into rows containing equal numbers of students, what possible arrangements can she have? 21 multiple choice Tilly Tyler has 24 green bathroom tiles left over. If she wants to use them all on the wall behind the kitchen sink (without breaking any) which of the following arrangements would be suitable? I 4 rows of 8 tiles II 2 rows of 12 tiles III 4 rows of 6 tiles IV 6 rows of 5 tiles V 3 rows of 8 tiles A I and II B I, II and III C III, IV and V D II, IV and V E II, III and V 22 Lisa needs to cut tubing into the largest pieces of equal length that she can without having any offcuts left over. She has 3 sections of tubing; one of length 6 metres, another of length 9 metres and the third of length 15 metres. a How long should each piece of tubing be? b How many pieces of tubing will Lisa end up with?

GAM

me E ti

Multiples, factors and primes 02

23 Mario, Luigi, Dee Kong and Frogger are playing Nintendo. Mario takes 2 minutes to play a complete game, Luigi takes 3 minutes, Dee Kong takes 4 minutes and Frogger takes 5 minutes. They have 12 minutes to play. a If they play continuously, which of the players would be in the middle of a game as time ran out? b After how many minutes did this player begin the last game?

51

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Chapter 2 Multiples, factors and primes

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1 What number am I? I am a multiple of 5 with factors of 6, 4 and 3. The sum of my digits is 6. 2 My age is a multiple of 3 and a factor of 60. The sum of my digits is 3. How old am I? (There are two possible answers.) 3 Find the highest common factor of 462, 504 and 630.

How many tiles? At the start of this chapter, you may have considered the following situation. A room measures 550 centimetres by 325 centimetres. What would be the side length of the largest square tile that can be used to tile the floor without any cutting?

1. 2. 3. 4.

Try this question now. (Hint: Find the common factors of 550 and 325.) How many tiles would fit on the floor along the wall of length 550 centimetres? How many tiles would fit on the floor along the wall of length 325 centimetres? How many floor tiles would be needed for this room?

52 Desertt creatur Deser creatures es that are are sheltered sheltered by by spinife spinif ex in outback Australia Maths Quest 7 for Victoria

The highest common factor (HCF) of the pair of numbers given and the letter beside each gives the puzzle answer code.

A

20, 50

D

15, 25

L

42, 56

O

18, 30

S

12, 21

R

40, 16

T

70, 4

N

36, 54

P

27, 45

I

21, 35

M

60, 80

F

120, 90

H

13, 7

C

36, 24

E

30, 45

G

39, 26

B

12, 20

Z

75, 200

14

15

13

14

15

3

3

14

7

25

10

8

5

3

9

7

15

5

7

3

1

4

15

15

2

14

15

3

15

10

5

15

12 8

3 15

15 3

2

8

2

15 5

5

13 8 10 3

2 3

1

10

1

6

5 9

5 9 15

15 8

8 3

1 6 9 9 7 18 13 20 7 12 15 5 8 10 13 6 18 30 14 7 15 3 9 10 7 18 2 15 5 30 7 8 15 2 10 7 14 30 7 18 12 1 15 3

Chapter 2 Multiples, factors and primes

53

Prime numbers A prime number is a counting number which has exactly 2 factors, itself and 1. The number 3 is a prime number. Its only factors are 3 and 1. The number 7 is a prime number. Its only factors are 7 and 1. A composite number is one which has more than two factors. The number 10 is a composite number; its factors are 1, 10, 5 and 2. The number 16 is also a composite number; its factors are 1, 16, 2, 8 and 4. The number 1 is a special number. It is neither a prime number nor a composite number because it has only one factor, 1.

WORKED Example 8 Find 3 prime numbers that are greater than 10. THINK 1

2 3 4 5 6

WRITE

All even numbers except 2 are composite numbers because factors include 1, 2 and the number itself. Look at odd numbers only. Try 11. Factors of 11 are 11 and 1 only, so it is a prime number. Try 13. Factors of 13 are 1 and 13 only, so 13 is a prime number. Try 15. Factors are 1, 3, 5, 15. Not a prime number. Try 17. Factors are 1 and 17 only. Prime number. List the 3 prime numbers found.

11 13

17 Three prime numbers greater than 10 are 11, 13 and 17.

Sum of two prime numbers Every composite number which is odd, can be written as the sum of 2 prime numbers, one of which is 2. For example,

9 = composite

7 prime

+

2 prime

Other examples are: 15 = 13 + 2 21 = 19 + 2 25 = 23 + 2 A mathematician, Goldbach, believed that every even composite number could be written as the sum of two odd prime numbers. For example, 8 = 5 + 3 24 = 5 + 19 46 = 17 + 29 Is he right? Can you find an even composite number which is not the sum of two prime numbers?

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Maths Quest 7 for Victoria

WORKED Example 9 Write each of the following as the sum of two odd prime numbers. a 14 b 28 THINK

WRITE

a

a 5 + 9 = 14 9 is not prime. 3 + 11 = 14 3 and 11 are prime.

Try a pair of odd numbers and check that they are prime. If they are not prime numbers, try other pairs of odd numbers until you have found a pair which are prime.

1 2

b Try a pair of odd numbers and check that they are prime.

b 5 + 23 = 28 5 and 23 are prime.

Often there is more than one solution. For 28, 5 + 23 = 28 and 11 + 17 = 28. The numbers 5, 23, 11 and 17 are all prime. Note that if the number is even, 2 will not be one of the prime numbers in the sum.

remember remember 1. A prime number is a counting number which has exactly two factors, itself and 1. 2. A composite number is one which has more than two factors. 3. The number one is not a prime number or a composite number.

2C

Prime numbers

1 a List the factors for each number between 1 and 20. b Use the factor lists to write down all the prime numbers up to 20. WORKED

Example Prime 8 numbers

2.3

2 Find 4 prime numbers which are between 20 and 40. 3 Can you find 4 prime numbers that are even? Explain.

4 Write each of the following as the sum of 2 odd prime numbers. a 8=_+_ b 12 = _ + _ 9 c 30 = _ + _ d 16 = _ + _ e 100 = _ + _ f 48 = _ + _ g 24 = _ + _ h 52 = _ + _ i 60 = _ + _ j 32 = _ + _

WORKED

Example

5 Answer true (T) or false (F) for each of the following. a All odd numbers are prime numbers. b No even numbers are prime numbers. c 1, 2, 3 and 5 are the first 4 prime numbers. d A prime number has 2 factors only.

Chapter 2 Multiples, factors and primes

55

6 multiple choice a The number of primes less than 10 is: A 4 B 3 C 5 D 2 E 6 b The first 3 prime numbers are: A 1, 2, 3 B 1, 3, 5 C 2, 3, 4 D 2, 3, 5 E 3, 5, 7 c The number 15 can be written as the sum of 2 prime numbers. These are: A 3 + 12 B 1 + 14 C 13 + 2 D 7+8 E 11 + 4 d Factors of 12 that are prime numbers are: A 1, 2, 3, 4 B 2, 3, 6 C 2, 3 D 2, 4, 6, 12 E 1 7 Twin primes are pairs of primes which are separated from each other by one even number. For example, 3 and 5 are twin primes. Find 2 more pairs of twin primes. 8 a Which of the numbers 2, 3, 4, 5, 6 and 7 cannot be the difference between 2 consecutive prime numbers? Explain. b For each of the numbers which can be a difference between 2 consecutive primes, give an example of a pair of primes less than 100 with such a difference.

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9 The following numbers are not primes. Each of them is the product of 2 primes. Find the 2 primes in each case. a 365 b 187

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1 What is the largest 3 digit prime number in which each digit is a prime number? 2 Find a prime number greater that 10 where the sum of the digits equals 11. 3 My age is a prime number. I am older than 50. The sum of my digits is also a prime number. If you add a multiple of 13 to my age the result is 100. How old am I? 4 I am a 3 digit number. I am divisible by 6. My middle digit is a prime number. The sum of my digits is 9. I am between 400 and 500. My digits are in descending order. 5 Angus is the youngest in his family and today he and his Dad share a birthday. Both their ages are prime numbers. Angus’s age has the same 2 digits as his Dad but in reverse order. In 10 years’ time, Dad will be three times as old as Angus. How old will each person be when this happens? 6 What is the largest 5-digit number you can write if each digit must be different and no digit may be prime?

2.4

2.1

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Maths Quest 7 for Victoria

History of mathematics E R ATOS T H E N E S O F C Y R E N E ( c 2 7 0 B C t o 1 9 0 B C ) During his life . . . The Roman Empire introduces a minted coin, the Denarius. Hannibal crosses the Alps. The Great Wall of China is built. Terracotta Soldiers are buried in a tomb at Xian (China). Eratosthenes was a Greek astronomer and mathematician. He was born in Cyrene (now Shahhat, Libya) in approximately 270 BC. Some sources suggest he was born in 270 BC and died in 194 BC. Eratosthenes was first educated by his father, Aglaos, and was then educated in Athens. He eventually became the main librarian at the Alexandrian museum. Eratosthenes was the first person to work out an accurate measurement of the circumference of the Earth (when many people believed the Earth was flat!). He did this by comparing the angle of the shadows of sticks at two distant locations, Alexandria and Athens, at the same time of the day.

Eratosthenes was one of the most famous mathematicians of his day and was asked by the ruler of Egypt, Ptolemy III, to tutor his son Ptolemy Philadelphus. He also took on the job of being the chief librarian at the University of Alexandra where other mathematicians, such as Euclid, had held this position. As chief librarian he was able to borrow and copy books from the library.

Because he worked on many types of research, some people thought that he was not a specialist in any area. However, all the work that he did was always of the highest standard. Eratosthenes made many discoveries in many areas that still benefit us now. He tried to improve the accuracy of calendars and realised there was a need for leap years to keep calendars in time with the seasons. He published the first map of the world based on longitude and latitude. He compiled a star catalogue containing 675 stars. Eratosthenes discovered a way of finding prime numbers, called the Sieve of Eratosthenes (see page 57). The largest known prime is 21 398 269 − 1 which has 420 921 digits, but new primes are now being discovered regularly with the help of computers. When Eratosthenes was old he became blind. He was so upset by this disability that he committed suicide.

Questions 1. What was Eratosthenes’s main job in life? 2. What physical property was he the first person to measure accurately? 3. What was different about Eratosthenes’s map of the world? 4. Which ruler did he work for? 5. What is the largest known prime at the time of this publication? 6. Why did Eratosthenes become depressed and commit suicide? Research 1. Look on the Internet for information about primes and find the largest prime number known at present and compare it with the one reported above. How much bigger is it now? 2. Why do we need leap years (they occur once every 4 years normally) and what is the difference between the years 1900 and 2000 in terms of leap years?

Chapter 2 Multiples, factors and primes

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The sieve of Eratosthenes An easy way to find prime numbers is to use the ‘Sieve of Eratosthenes’. Eratosthenes discovered a simple method of sifting out all of the composite numbers so that only prime numbers are left. You can follow the steps below to find all prime numbers between 1 and 100. Alternatively you can use the Excel file provided on the Maths Quest CD-ROM to simulate this process. Sieve of Eratosthenes (a) Copy the numbers from 1 to 100 in a grid as shown below. Use 1 centimetre square grid paper.

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

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

65

66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

(b) Cross out 1 as shown. It is not a prime number. (c) Circle the first prime number, 2. Then cross out all of the multiples of 2. (d) Circle the next prime number, 3. Now cross out all of the multiples of 3 that have not already been crossed out. (e) The number 4 is already crossed out. Circle the next prime number, 5. Cross out all of the multiples of 5 that are not already crossed out. (f) The next number that is not crossed out is 7. Circle 7 and cross out all of the multiples of 7 that are not already crossed out. Continue until you have circled all of the prime numbers and crossed out all of the composite numbers.

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Maths Quest 7 for Victoria

1. 2. 3. 4. 5.

Answer the following questions using the Sieve of Eratosthenes. How many prime numbers are there between 1 and 100? What is the largest prime number less than 100? How many prime numbers are there between 20 and 100? How many single digit prime numbers are there between 1 and 100? How many double digit prime numbers are there between 1 and 100?

1 1 Find all the factors of 64. 2 List the factor pairs of 24. 3 Find the highest common factor of 30 and 45. 4 List the first 5 multiples of 9. 5 Find the lowest common multiple of 4 and 9. 6 Find the lowest common multiple of 12 and 16. 7 List all the prime numbers between 20 and 50. 8 Write 20 as the sum of 2 prime numbers. 9 Write 50 as the sum of 2 prime numbers. 10 A school marching band is made up of 36 students. The band leader is trying to arrange them into rows containing equal numbers of students. What arrangements are possible?

Prime factors and factor trees A factor tree shows the prime factors of a composite 20 number. Each branch shows a factor of all numbers above it. Factors of 20 2 10 The last numbers are all prime numbers, therefore they are prime factors of the original number. From the factor tree shown, 2, 2 and 5 are prime factors Factors of 10 2 5 of 20. Note that the factor 2 need only be written once, so we 20 can write that the prime factors of 20 are 2 and 5. If we had chosen different factors of 20 to start with, would 4 5 we end up with different prime factors? Choosing 4 and 5 instead of 10 and 2 did not change the prime factors of 20. If all the prime factors are multiplied together, 2 2 the answer will be the original number. Prime factors are 2 and 5 2 × 2 × 5 = 20

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Chapter 2 Multiples, factors and primes

WORKED Example 10

a Find the prime factors of 50 by drawing a factor tree. b Write 50 as a product of its prime factors. THINK

WRITE a

a

1

Find a factor pair of the given number and begin the factor tree (50 = 5 × 10).

2

If a branch is prime, no other factors can be found (5 is prime). If a branch is composite, find factors of that number; 10 is composite so 10 = 5 × 2. Continue until all branches end in a prime number then stop. Write the prime factors.

50

5

3 4

b Write 50 as a product of prime factors found in part (a).

10 50

5

10

5

2

2 and 5 are prime factors of 50. b 50 = 5 × 5 × 2

WORKED Example 11 Find the prime factors of 56. THINK 1

WRITE

Draw a factor tree. When all factors are prime numbers you have found the prime factors.

56

8

2

4

2 2

Write the prime factors.

7

2

The prime factors of 56 are 7 and 2.

remember remember 1. A factor tree shows the prime factors of a composite number. 2. The last numbers in the factor tree are all prime numbers, therefore they are prime factors of the original number. 3. Every composite number can be written as a product of prime factors. For example, 20 = 2 × 2 × 5.

20

4

2

5

2

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Maths Quest 7 for Victoria

2D WORKED

Example

Prime factors

Prime factors

1

Prime factors and factor trees

i Find the prime factors of each of the following numbers by drawing a factor tree. ii Write each one as a product of its prime factors.

10

a d g j 2

b e h k

15 100 18 84

c f i l

30 49 56 98

24 72 45 112

i Find the prime factors of the following numbers by drawing a factor tree. ii Express the number as a product of its prime factors. a d g j

Prime factors

b e h k

40 121 3000 196

c f i l

35 110 64 90

32 150 96 75

3 multiple choice a A factor tree for 21 is: A 21

7

D

B

3

C

21

1

21

3

21

E

b A factor tree for 36 is: A 36

1

B

36

1

7

1

2

C

18

36

9

4

36

9

9

7

36

18

2

3

E

36

2

c

7

7×1 1

D

1

21

3 3×1

21

3

4

3

The prime factors of 16 are: A 1, 2 B 1, 2, 4 C 2, 4, 8

d The prime factors of 28 are: A 1, 28 B 2, 7 C 1, 2, 14

2

2

D 2

E 1, 2, 4, 8, 16

D 1, 2, 7

E 2, 7, 14

Chapter 2 Multiples, factors and primes

4 Find the prime factors of each of the following numbers. a 48 b 200 11 d 81 e 18 g 27 h 300 j 120 k 50

61

WORKED

Example

c f i l

42 39 60 80

QUEST

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MAT H

5 State whether each of the following is true (T) or false (F). a The number 3 is the only prime factor of 9. b No two numbers can have the same prime factors. c The numbers 2, 3, 5 and 7 are the prime factors of 210. d The numbers 1, 2 and 5 are the prime factors of 40. e The prime factors of 220 are 2, 5 and 11. f All numbers have exactly 2 prime factors.

CH

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1 A whole number is ‘perfect’ if it equals the sum of all its proper factors. The proper factors of a number are all factors smaller than the number. So 6 is perfect since its proper factors are 1, 2 and 3 and 6 = 1 + 2 + 3. a Find all the proper factors of 28 and show that 28 is a perfect number. b Do the same for 496. 2 Find the 7 proper factors of the number 999. Is 999 a perfect number? 3 Use a calculator to find the 13 proper factors of 8128. Is 8128 a perfect number?

Index notation The product of factors can be written in a shorter form by using index notation or index form. A number in index form has two parts, the base and the index. The product of factors 3 × 3, can be written as 32. The 3 is the base and the 2 is the index. Two other examples are given below: 4 × 4 × 4 = 43 43 has a base of 4 and an index of 3. 5 2×2×2×2×2=2 25 is a number in which 2 is the base and 5 is the index. 5 2 is in index notation or index form. 2 × 2 × 2 × 2 × 2 is in expanded form. A composite number written as a product of prime factors can be written using index notation. So 50 = 2 × 5 × 5 = 2 × 52 and 56 = 2 × 2 × 2 × 7 = 23 × 7.

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Maths Quest 7 for Victoria

WORKED Example 12 Write the following using index notation. a 5×5 b 2×2×6×6×6 THINK

WRITE

a

Write the multiplication. Write the number being multiplied as the base and the number of times it is multiplied as the index.

a 5×5 = 52

Write the multiplication. Write the number being multiplied as the base and the number of times it is multiplied as the index.

b 2×2×6×6×6 = 2 2 × 63

1 2

b

1 2

WORKED Example 13 Write 120 as a product of prime factors using index notation. THINK 1

WRITE

Find a factor pair and begin the factor tree. If the number on the branch is a prime number, stop. If not, continue until a prime number is reached.

120

12

4

2 2 3

Write the number as a product of prime factors. Write your answer using index notation.

10

3

5

2

120 = 2 × 2 × 2 × 3 × 5 120 = 23 × 3 × 5

WORKED Example 14

Write 53 in expanded form and then find the answer. THINK 1 2 3

4

Write the question in expanded form. Multiply the first 2 numbers. Multiply the answer by the next number and continue until all numbers have been multiplied. Write the answer.

2

WRITE 53 = 5 × 5 × 5 = 25 × 5 = 125 53 = 125

Chapter 2 Multiples, factors and primes

Graphics Calculator tip!

63

Calculating numbers in index form

To calculate a number in index form, first type in the base, then ^ , then the index and press ENTER . The screen opposite shows the calculation for 53 (from worked example 14).

remember remember 1. The product of prime factors can be written in a shorter form by using index notation. For example, 40 = 23 × 5. Using index notation, 23 is a number with base 2 and index of 3. 2. The 2 (or base number) is the number being multiplied and the 3 (or index) indicates how many times the base is being multiplied 23 = 2 × 2 × 2.

2E WORKED

Example

12a

WORKED

Example

12b

WORKED

Example

13 WORKED

Example

14

Index notation

1 Write the following using index notation. a 4×4×4 b 8×8 c 7×7×7×7 d 12 × 12 × 12 × 12 × 12 e 2×2×2×2×2×2×2 f 13 × 13 × 13 g 5×5×5 h 9×9×9×9×9×9×9×9 2 Write the following using index notation. a 2×2×3 b 3×3×3×3×2×2 c 5×5×2×2×2×2 d 7×2×2×2 e 5 × 11 × 11 × 3 × 3 × 3 f 13 × 5 × 5 × 5 × 7 × 7 g 2×2×2×3×3×5 h 3×3×2×2×5×5×5 3 Write the following as a product of prime factors using index notation. a 60 b 50 c 75 d 220 e 192 f 72 g 124 h 200 4 Write the following in expanded form and then find the answer. b 112 c 53 d 24 a 32 2 6 3 2 e 2×7 f 3×2 g 9 ×3 h 2 5 × 42 4 3 2 2 3 i 3 +7 j 2 +3 k 10 − 3 l 5 3 − 24 4 2 5 3 4 2 m 2 ÷2 n 3 ÷ 3 (Hint: Write 2 ÷ 2 as a fraction and simplify.) 5 Write one million using index notation. Use 10 as the base number. 6 multiple choice The largest number listed here is: A 150 B 26 C 34

D 53

E 72

7 multiple choice The smallest number listed here is: B 102 C 115 A 43

D 025

E 44

Index notation

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Maths Quest 7 for Victoria

Divisibility Often it can be difficult to find all the factors of a number, or to work out if a number is prime or composite. The following tests will help you.

Divisibility tests — 2, 3 and 4 A number is divisible by a second number if it can be divided by the second number without a remainder. For example, 20 is divisible by 10, because 20 can be divided by 10, without a remainder; that is, 20 ÷ 10 = 2. There is a way to test whether a number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10 or 11.

Divisible by 2 The number 2 goes into all even numbers. A number is divisible by 2 if it is even. Even numbers end in 0, 2, 4, 6 or 8. Divisible by 3 All of the numbers in the 3 times table listed below are divisible by 3. 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39 . . . The sum of the digits in each number is 3, 6, 9, 3, 6, 9, 3, 6, 9, 3, 6, 9, 12 . . . Each of these sums is also divisible by 3. If the sum of the digits of a number is divisible by 3, then that number is also divisible by 3.

WORKED Example 15 State whether each of the following numbers is divisible by 3. a 45 b 92 c 66 d 5742 e 1 233 693 THINK To be divisible by 3 the sum of the digits must be divisible by 3. a For 45, the sum of digits is 4 + 5 = 9. Sum is divisible by 3.

WRITE

a 45 is divisible by 3.

b For 92, the sum of digits is 9 + 2 = 11. Sum is not divisible by 3.

b 92 is not divisible by 3.

c For 66, the sum of digits is 6 + 6 = 12. Sum is divisible by 3.

c 66 is divisible by 3.

d For 5742, the sum of digits is 5 + 7 + 4 + 2 = 18. Sum is divisible by 3.

d 5742 is divisible by 3.

e For 1 233 693, the sum of digits is 1 + 2 + 3 + 3 + 6 + 9 + 3 = 27. Sum is divisible by 3 (because 2 + 7 = 9 which is divisible by 3).

e 1 233 693 is divisible by 3.

Chapter 2 Multiples, factors and primes

65

Divisible by 4 If the last two digits of a number are divisible by 4, then the number is divisible by 4. The number 2032 is divisible by 4 because the last 2 digits, 32, are divisible by 4 (32 ÷ 4 = 8).

WORKED Example 16 State whether each of the following numbers is divisible by 4. a 48 b 3012 c 150 THINK

WRITE

Consider the last 2 digits and check whether this number is divisible by 4. a Last two digits are 48 which is divisible by 4 (48 ÷ 4 = 12).

a 48 is divisible by 4.

b Last two digits are 12 which is divisible by 4 (12 ÷ 4 = 3).

b 3012 is divisible by 4.

c Last two digits are 50 which is not divisible by 4 (50 ÷ 4 = 12.5).

c 150 is not divisible by 4.

remember remember 1. Even numbers are divisible by 2. 2. If the sum of the digits of a number is divisible by 3, then that number is also divisible by 3. 3. If the last 2 digits of a number are divisible by 4, then the number is divisible by 4.

2F WORKED

Example

15

WORKED

Example

16

Divisibility tests — 2, 3 and 4

For questions 1 to 4 write ‘yes’ if divisible and ‘no’ if not divisible. 1 State whether each of the following numbers is divisible by 3. a 27 b 51 c 38 d 126 e 4017 f 943 g 362 h 84 i 69 j 9462 k 523 l 763 836 2 State whether each of the following is divisible by 2. a 38 b 17 c 26 e 288 f 100 g 370 i 2020 j 719 k 4533

d 3093 h 4131 l 64 512

3 State whether each of the following numbers is divisible by 4. a 44 b 28 c 328 e 917 f 3041 g 6084 i 149 j 70 232 k 69 478

d 212 h 68 l 324 636

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Maths Quest 7 for Victoria

4 State whether each of the following is divisible by 2, 3 and 4. a 12 b 30 c 48 e 180 f 236 g 3690 i 200 j 1116 k 92 472

d 312 h 552 l 75 148

5 Copy the following list and circle the numbers that are divisible by both 2 and 3. 87, 18, 108, 127, 12, 45, 3024, 96, 67, 429, 216 6 Copy the following list and circle the numbers that are divisible by 3 and 4. 12, 390, 420, 96, 880, 612, 264, 1038, 59, 2003 7 multiple choice A 250 gram block of Dairy Milk Chocolate has 60 pieces. It can be shared evenly between the following number of people: A 2, 5, 7

B 3, 5, 9

C 2, 5, 11

D 3, 5, 12

E 3, 10, 14

8 If there are 220 students in Year 7, could they form groups of: a 2 with no students left over? b 3 with no students left over? c 4 with no students left over? 9 A tennis team consists of 4 players. If a tennis club has 326 members who want to play in a team, will they be able to make teams of exactly 4 players, or will there be some players left over?

2.2 10 There are 294 students waiting to buy their lunch at the school canteen and there are 3 queues. Could the 3 queues be exactly the same size?

67

QUEST

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Chapter 2 Multiples, factors and primes

CH

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1 If you double my age and subtract 1, you will have a prime number. I am less than 20 years old. The sum of my digits is divisible by 2 and 3. How old am I? 2 Find the largest 8-digit number which is divisible by 2, 3 and 4. 3 A clock strikes at regular intervals. It strikes the number of hours on the hour, twice for every half hour and once for each of the remaining quarter hours. How many times does the clock strike in a 24-hour period?

2 1 List the first 3 multiples of 15. 2 List the factor pairs of 100. 3 Find the prime factors of 48 by drawing a factor tree. 4 Find the prime factors of 81. For questions 5 to 8, consider the following list of numbers: 36, 77, 108, 231, 459, 12 568. 5 Which numbers in the list are divisible by 2? 6 Which numbers in the list are divisible by 3? 7 Which numbers in the list are divisible by 4? 8 Which numbers in the list are divisible by 2, 3 and 4? 9 Jonathon has just won an enormous block of chocolate which is made up of 1344 little squares. Can he divide it evenly between himself and 3 friends? 10 At the swimming pool there are 3 water slides. On a really hot day there were 524 people queuing up for a turn. Is it possible that the 3 queues could be the same length?

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Maths Quest 7 for Victoria

Divisibility tests — 5, 6, 7 and 8 Divisible by 5 The 5 times table shows all the numbers that 5 ‘goes into’. These include 5, 10, 15, 20, 25, 30, 35, 40, … All of these numbers end in 5 or 0. Numbers ending in 5 or 0 are divisible by 5.

WORKED Example 17 State whether each of the following is divisible by 5. a 2457 b 6730 THINK

WRITE

If the number ends in 0 or 5 it is divisible by 5. a The number ends in 7, not 0 or 5.

a 2457 is not divisible by 5.

b The number ends in 0.

b 6730 is divisible by 5.

Divisible by 6 A number is divisible by 6, if it is divisible by both 2 and 3.

WORKED Example 18 Find whether 324 is divisible by 6. THINK 1 2

3

WRITE

Check whether 324 is even and therefore divisible by 2. Check whether the sum of the digits is divisible by 3 so that the number is divisible by 3. Write the answer.

324 is even, so it is divisible by 2. 3 + 4 + 2 = 9, so 324 is divisible by 3.

324 is divisible by 6.

Divisible by 7 To test whether a number is divisible by 7 is more complicated. The following steps can be followed. 1. Write the number without the last digit. 2. Subtract 2 times the last digit from this number. 3. Repeat steps 1 and 2 until you are left with a 1 or 2 digit number. 4. If the 1 or 2 digit number is divisible by 7, the original number is divisible by 7. If the number you are left with is 0, the original number is divisible by 7.

Chapter 2 Multiples, factors and primes

69

WORKED Example 19 State whether the following are divisible by 7. a 2483 b 1967 THINK

WRITE

a

Write the number without the last digit. Subtract 2 times the last digit (2 × 3 = 6) from the number obtained in step 1. Repeat the process in order to obtain a 1 or 2 digit number. Write the number without the last digit. Subtract 2 times the last digit (2 × 2 = 4) from the number obtained in step 3. Check whether the 2 digit number obtained is divisible by 7. Write the answer.

a 248 248 − 6 = 242

Write the number without the last digit. Subtract 2 times the last digit (2 × 7 = 14). Repeat. Write without the last digit. Subtract 2 times the last digit (2 × 2 = 4). Check whether the 2-digit number is divisible by 7. Write the answer.

b 196 196 − 14 = 182 18 18 − 4 = 14 14 is divisible by 7.

1 2 3

4 5 6

b

1 2 3 4 5 6

24 24 − 4 = 20 20 is not divisible by 7. So 2483 is not divisible by 7.

So 1967 is divisible by 7.

Divisible by 8 If the last 3 digits of a number are divisible by 8 then the number is divisible by 8. The number 1032 is divisible by 8 because 032 is divisible by 8. The number 2804 is not divisible by 8 because 804 is not divisible by 8.

WORKED Example 20 State whether the following are divisible by both 5 and 6. a 270 b 4305 THINK

WRITE

a

a 270 ends in 0 so it is divisible by 5.

1 2

3

Check whether the number is divisible by 5. Check whether the number is divisible by 6 by checking if it is divisible by both 2 and 3. If the number is even, it is divisible by 2. If the sum of the digits is divisible by 3, the number is divisible by 3. Write the answer.

270 is even so it is divisible by 2. 2 + 7 + 0 = 9 which is divisible by 3. So 270 is divisible by 6.

270 is divisible by both 5 and 6.

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Maths Quest 7 for Victoria

THINK

WRITE

b

b 4305 ends in 5 so it is divisible by 5.

1 2 3

Check whether the number is divisible by 5. Check whether the number is divisible by 6. Write the answer.

4305 is not divisible by 2 and so it is not divisible by 6. 4305 is not divisible by both 5 and 6.

remember remember 1. Numbers ending in 0 or 5 are divisible by 5. 2. Numbers which are divisible by both 2 and 3 are divisible by 6. 3. To test whether a number is divisible by 7, drop the last digit, and subtract twice the dropped digit from the others. Repeat this process until there are only 2 digits left and see if that number is divisible by 7. 4. If the last 3 digits are divisible by 8 then the number is divisible by 8.

2G

Divisibility tests — 5, 6, 7 and 8

For questions 1 to 5 write ‘yes’ if divisible and ‘no’ if not divisible. WORKED 1 State whether each of the following is divisible by 5. Example a 5 b 30 c 47 d 105 e 5032 17 f 1175 g 943 h 77 i 39 260 j 645 2 State whether each of the following is divisible by 6. a 42 b 63 c 54 d 124 18 f 312 g 4311 h 3106 i 9 246 000

e 96 j 72 144

3 State whether each of the following is divisible by 7. a 483 b 923 c 658 d 4235 19 f 1855 g 6007 h 9948 i 38 297

e 3126 j 64 582

WORKED

Example

WORKED

Example

4 State whether each of the following is divisible by 8. a 64 b 45 c 4064 d 4008 f 1020 g 112 h 55 112 i 42 391 016 5 State whether each of the following is divisible by both 5 and 6. a 30 b 250 c 18 000 d 630 20 f 2952 g 2040 h 960 i 197 460

e 71 045 j 3102

WORKED

Example

e 375 j 3 416 220

6 Copy the following list and circle the numbers that are divisible by both 5 and 8. 4040, 100, 240, 6024, 400, 10 000, 367 080

Chapter 2 Multiples, factors and primes

71

7 State true (T) or false (F) for each of the following. a 148 is divisible by 6. b 49 026 is divisible by 5. c 810 560 is divisible by 8. d 3510 is divisible by both 5 and 6. e 6080 is divisible by both 5 and 8. f 21 960 is divisible by both 6 and 8. g 410 is divisible by both 5 and 6. h 13 160 is divisible by both 6 and 8.

QUEST

GE

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MAT H

8 Could 142 students be broken up into volleyball teams, with no one left out, if each team has 6 players exactly?

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1 How old am I? I am older than 9. The sum of my digits is 3. My age is divisible by 5 and 2. 2 My age is a multiple of the sum of the first 2 prime numbers. It is also divisible by 7 and has exactly 4 factors. How old am I? 3 My locker number has factors of 2, 5 and 9. Find my locker number if there are 9 other factors of this number. 4 What are the two numbers that are divisible by both 5 and 7 and have exactly 12 factors?

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Maths Quest 7 for Victoria

Divisibility tests — 9, 10 and 11 Divisible by 9 A number is divisible by 9 if the sum of its digits is divisible by 9.

WORKED Example 21 State whether each of the following is divisible by 9. a 360 b 296 THINK

WRITE

a

Add the digits of the number. Check whether the sum of the digits is divisible by 9.

a 3 + 6 + 0 = 9. 9 is divisible by 9 so 360 is divisible by 9.

Add the digits of the number. Check whether the sum of the digits is divisible by 9.

b 2 + 9 + 6 = 17 The number 17 is not divisible by 9 so 296 is not divisible by 9.

1 2

b

1 2

Divisible by 10 The 10 times table shows all of the numbers 10 ‘goes into’. 10, 20, 30, 40, 50, . . . All of these numbers end in 0. Numbers ending in zero are divisible by 10. Divisible by 11 Divisibility by 11 is more complicated. To test whether a number is divisible by 11, the position of each digit must be considered as shown. For the number 6542, the 6 is in an odd position because it is in position 1 (1 is odd). The number 5 is in an even position because it is in position 2 (2 is even). The number 4 is in an odd position because it is in position 3 (3 is odd). The number 2 is in an even position because it is in position 4 (4 is even). Number 6 5 4 2 ↕ ↕ ↕ ↕ Position 1 2 3 4 If the sum of the odd-positioned digits equals the sum of the even-positioned digits or differs from the sum of the even-positioned digits by 11, then the number is divisible by 11. For example, 55 is divisible by 11 because 5 = 5. 6542 is not divisible by 11 because 6 + 4 ≠ 5 + 2 1364 is divisible by 11 because 1 + 6 = 3 + 4. 24 968 is not divisible by 11 because 2 + 9 + 8 ≠ 4 + 6. 913 is divisible by 11 because 9 + 3 ≠ 1 but the two sums differ by 11.

Chapter 2 Multiples, factors and primes

73

WORKED Example 22 State whether 495 is divisible by both 9 and 11. THINK 1

2

3

WRITE

The number is divisible by 9 if the sum of the digits is divisible by 9 (18 ÷ 9 = 2). The number is divisible by 11 if the sum of the odd-positioned digits (4 + 5) is equal to the sum of the evenpositioned digits (9). Write the answer.

4 + 9 + 5 = 18 so 495 is divisible by 9. 4 + 5 = 9 so 495 is divisible by 11.

495 is divisible by both 9 and 11.

Finding prime factors The divisibility tests can be used to find the prime factors of any number.

WORKED Example 23 Find the prime factors of 2160. THINK 1

2

3

4

WRITE

The number 2160 is divisible by 10 because 2160 ends in 0. Use 10 as a factor in the factor tree. The number 216 is divisible by 9 because 2 + 1 + 6 = 9 and 9 is divisible by 9. Use 9 as a factor in the factor tree. The number 24 is divisible by 3 because 2 + 4 = 6 and 6 is divisible by 3. Use 3 as a factor. The number 8 is even so use 2 as a factor and continue until all prime factors are found.

2160

10

5

216

9

2 3

24 3

3

8 2

4 2

5

Write the number as a product of prime factors using index notation.

2

2160 = 24 × 33 × 5

remember remember 1. A number is divisible by 9 if the sum of the digits is equal to 9. 2. A number is divisible by 10 if the number ends in 0. 3. A number is divisible by 11 if the sum of the odd-placed digits is equal to the sum of the even-placed digits or the two sums differ by 11.

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Maths Quest 7 for Victoria

2H

Divisibility tests — 9, 10 and 11

For questions 1 to 3, write ‘yes’ if divisible and ‘no’ if not divisible. 1 State whether each of the following is divisible by 9. Example a 81 b 108 c 362 d 261 21 e 918 f 1602 g 34 901 h 60 201 i 7096 j 8 446 520 k 356 892 l 459 258 WORKED

WORKED

Example

2 State whether each of the following is divisible by 10. a 30 b 45 c 600 e 891 f 8910 g 365 i 77 602 j 714 980 k 1245

d 750 h 4120 l 3 205 400

3 Find whether each of the following is divisible by 11. a 44 b 99 c 132 e 931 f 721 g 241 i 6822 j 50 260 k 4 281 926

d 594 h 3102 l 8277

4 Copy the following list and circle the numbers that are divisible by both 9 and 11.

22

45, 99, 990, 1010, 32 313, 198, 297, 2970, 5167 5 State true (T) or false (F) for each of the following. a 628 010 is divisible by 10. b 36 256 is divisible by 11. c 2107 is divisible by 9. d 13 068 is divisible by 9 and 11. e 79 010 is divisible by 9 and 11. f 332 132 130 is divisible by 9, 10 and 11. g 349 160 is divisible by 9, 10 and 11. h 5 832 910 is divisible by 9, 10 and 11.

6 Write the following numbers as the product of prime factors using index notation. Use divisibility tests to help. 23 a 1575 b 616 c 1080 d 2646

WORKED

Example

7 Use a divisibility test to show that the following numbers are composite. a 7667 b 9229

2.3

QUEST

GE

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MAT H

8 Use divisibility tests to decide which of the following numbers are primes. a 119 b 209 c 103 d 91

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1 My age is a 2 digit number. It is divisible by 2, 3, 6 and 9. I am older than 20 but younger than 50 years old. How old am I? 2 My age is greater than 10 and less than 20. It is a composite number that is divisible by 3 and 9. How old am I? 3 I am very old. My age is a 3 digit number that is divisible by 11. It is also a multiple of 4. How old am I? 4 My age is divisible by 11. It is also a multiple of 2 and has 6 factors. I am less than 50 years old. How old am I?

Chapter 2 Multiples, factors and primes

75

Square numbers and square roots Square numbers If a number has 2 equal factors it is called a square number. Some examples are given below: 4 is a square number because 2 × 2 = 4 9 is also a square number because 3 × 3 = 9 1 is also a square number because 1 × 1 = 1. The number 1 is the first square number, 4 is the second square number, 9 is the third square number . . . 1 = 1 × 1 = 12

4 = 2 × 2 = 22

9 = 3 × 3 = 32

Each of these numbers can be represented by dots in the following way. Can you see why they are called square numbers?

12 = 1

22 = 4

32 = 9

WORKED Example 24 Find the sixth square number. THINK 1

2 3

Write the number which shows the square number you are looking for. Multiply that number by itself. Find the answer. Write your answer in a sentence.

WRITE 6×6 = 36 So 36 is the sixth square number.

WORKED Example 25 Find the square numbers between 90 and 150. THINK 1 2

3

Use your knowledge of tables to find the first square number after 90. Find the square numbers which come after that one but before 150. Write the answer in a sentence.

WRITE 102 = 10 × 10 = 100 112 = 11 × 11 = 121 122 = 12 × 12 = 144 132 = 13 × 13 = 169 (too big) The square numbers between 90 and 150 are 100, 121 and 144.

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Maths Quest 7 for Victoria

Graphics Calculator tip!

Calculating the square of a number

There are 3 ways to obtain the square of a number using a graphics calculator. 1. Type in the base, press ^ , then type in the index which is 2. Press ENTER to complete the calculation. x 2 , then 2. Type in the base, press the key marked press ENTER . 3. Type in the number, press the × sign then type in the same number again. Remember to press ENTER . All 3 methods are shown in the screen opposite.

Square roots Finding the square root of a number is the opposite of squaring a number. 42 = 16 means that 4 multiplied by itself is equal to 16. 16 = 4 means that we are finding a number that multiplies by itself to equal 16.

WORKED Example 26

Find: a

49

b

3600 .

THINK

WRITE

a Find a number which when multiplied by itself is equal to 49.

a 49 = 7 × 7 so

b Find a number which when multiplied by itself is equal to 3600. You may need to try a few different numbers until you find the right one.

b

Graphics Calculator tip!

So

49 = 7

3600 = 60 × 60 3600 = 60

Calculating the square root of a number

To calculate the square root of a number, first press 2nd [ ] then the number. Remember to press ENTER to complete the calculation. The operation x2 . is listed in yellow above the key marked This is why we need to first press the key marked 2nd . The screen shows the calculations needed for worked example 26.

remember remember 1. A square number is one which has 2 equal factors. 2. Finding the square root of a number is the opposite of squaring a number. 3. To find the square root of a number, find a number that multiplies by itself to equal the original number.

Chapter 2 Multiples, factors and primes

2I WORKED

Example

24 WORKED

Example

25

77

Square numbers and square roots

1 Find the eighth square number. 2 Find the following square numbers: a the 13th b the 15th c the 20th

d the 50th

e the 100th. Square numbers

3 a Find the square numbers between 50 and 100. b Find the square numbers between 160 and 200. 4 Find the even square numbers between 10 and 70.

2.3

5 Find the odd square numbers between 50 and 120. 6 a Find the prime numbers between 12 and 22. b Find the prime numbers between 62 and 72. 7 a b c d

WORKED

Example

26a WORKED

Example

26b

Find the sum of the first 2 odd numbers. Find the sum of the first 3 odd numbers. Find the sum of the first 4 odd numbers. The answers to a, b and c are all square numbers. Use this to find the sum of i the first 10 odd numbers ii the first 50 odd numbers iii the first 1000 odd numbers.

8 Find: 25 a 9 Find: 4900 a

b

64

b

14 400

c

b 92 − 36 e 3 2 − 22 ÷ 4 +

d

360 000

49

c f

160 000

52 × 22 × 49 9 × 42 − 144 ÷ 22

GE

QUEST

EN

MAT H

d

121

Square roots

10 Simplify the following. a 22 + 25 d 32 + 22 × 16

S

c

81

CH

AL

L

1 Megan has 3 game scores which happen to be square numbers. The first 2 scores have the same 3 digits. The total of the 3 scores is 590. What are the 3 scores? 2 The difference of the squares of two consecutive odd numbers is 32. What are the two odd numbers? 3 What is the number? It is greater than 152. It is not a multiple of 3 or 5. It is less than 202. It is not a multiple of 17. It is a square number. 4 Find a 2-digit number in which both digits are different and the difference between the square of the number and the square of the number with the digits reversed is a square number.

2.5

78

Maths Quest 7 for Victoria

Why did the woman Why woman leav leave the theatre theatre after Act 1? Determine the exact values of the terms below and match them up with the letter beside each to find the answer code.

16

A

32 =

D

43 =

G

49 =

I

24 =

M

62 =

O

25 =

R

25 =

C

36 =

E

33 =

H

52 =

L

9=

N

72 =

P

23 =

S

53 =

T

92 =

W

22 =

X

125 27

“ 12

144 =

6

49

32 36

5

125 3

125

36 64 125 27

5

81 8

12 36 25

16

’

8 8

12

12

7 7

125

9 4

6

5

”

Chapter 2 Multiples, factors and primes

79

Testing whether numbers are prime or composite You will need a calculator for this activity. Eratosthenes devised a way of testing whether numbers were prime or composite. He tested all the primes smaller than the square root of the number. If any of the primes divide evenly into the number then the number is composite. If none of the primes divide evenly into the number then it is prime. Let’s try this method with the number 1973. First find the square root of 1973 using a calculator, 1973 = 44.418 . . . The prime numbers less than 44 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 and 41. Test each prime to see whether the number 1973 is divisible by that prime. None of the primes up to 44 divide evenly into 1973. So 1973 is a prime number. Try this method with the following numbers. Classify each as prime or composite: 1. 1001 2. 3133 3. 8191.

A prime number is born! You can form a number which is prime by following some steps involving squaring a number. Start with a whole number. Add this number to the square of the number then add 41. For example, if the number chosen is 1, then we obtain 12 + 1 + 41 = 43 which is a prime number. 1. What is the prime number formed when the number you start with is: (a) 2 (b) 3 (c) 4 (d) 5 (e) 11 (f) 34 2. Can you find a prime number, which is less than 50, as your starting number which does not give a prime number answer when you follow this process? You may like to check your answers by using the prepared Excel spreadsheet on the Maths Quest CD-ROM. Use this spreadsheet to answer the following questions. 3. What is the prime number formed when the number you start with is: (a) 85 (b) 123 (c) 237 4. Find the starting number which produces the largest prime number under 10 000 using this method.

Producing primes

80

Maths Quest 7 for Victoria

summary 1 2 3

4 5 6 7 8 9 10 11 12

Copy the sentences below. Fill in the gaps by choosing the correct word or expression from the word list below. A is a whole number that divides exactly into another whole number, with no remainder. Factors that are the same for 2 or more numbers are called . The largest of these is known as the (HCF). A is obtained by multiplying a number by another number. If two or more numbers have the same multiple, they are called common multiples. The smallest of these is known as the (LCM). A has two factors only, itself and 1. A has more than 2 factors. The number is neither a prime number nor a composite number. Composite numbers can be written as the product of factors. The product of factors can be written in a shorter form by using notation or index form. A number in index form can be written using notation. Numbers with 2 equal factors are called numbers. Taking the square root of a number is the opposite of a number. Tests of divisibility: A number is divisible by: 2 if it ends in an number. 3 if the sum of its digits is divisible by . 4 if the last digits are divisible by 4. 5 if it ends in 0 or . 6 if it is divisible by and 3. 8 if the last three digits are divisible by . 9 if the of the digits is divisible by 9. 10 if it ends in . 11 if the sum of the even-positioned digits equals the sum of the -positioned digits or the two sums differ by 11.

WORD

LIST

one prime number square squaring composite number 2 lowest common multiple even

common factors 3 highest common factor sum index two 8 5

multiple odd factor 0 prime expanded

Chapter 2 Multiples, factors and primes

81

CHAPTER review 1 List the first 5 multiples of each number. a 11 b 100 d 20 e 13

c f

5 35

2A

2 Find the lowest common multiple (LCM) of the following pairs of numbers. a 3 and 12 b 6 and 15 c 4 and 7 d 5 and 8

2A

3 In a race, one trail bike rider completes each lap in 40 seconds while another completes it in 60 seconds. How long after the start of the race will the 2 bikes pass the starting point together?

2A

4 Find all the factors of each of the following numbers. a 16 b 27 c 50 d 42 e 36 f 72

2B

5 Find the highest common factor of each of these pairs of numbers. a 8 and 20 b 15 and 35 c d 18 and 60 e 11 and 77 f

14 and 42 6 and 72

6 List the factor pairs of the following numbers. a 24 b 40 d 21 e 99

48 100

c f

2B 2B

7 Dhiba wants to cut equal lengths of streamers to decorate a hall. She wants them to be as long as possible. If she has a roll containing 15 metres and another containing 35 metres, what should be the length of each streamer to have no left over sections?

2B

8 List all of the prime numbers less than 30.

d 124

2C 2C 2C 2D

13 Write the following as prime factors using expanded form. a 44 b 132 c 150 d 360

2D 2E

9 How many single digit prime numbers are there? 10 Find the prime number which comes next after 50. 11 Find the prime factors of: a 99 b 63

c

125

12 Express 280 as a product of prime factors.

14 Write the following using index notation. a 2×2×2×2×2×2 b 5×3×3×7

c

2×2×2×3×3×5

2E

82

Maths Quest 7 for Victoria

15 Find the 2 smallest numbers which are divisible by both 2 and by 3.

2F 2F

16 State whether the following are divisible by both 3 and by 4. a 120 b 155 c 76 d 252

2F,G,H

17 a State the test of divisibility for: i 2 ii 5 iii 10. b What do these 3 tests have in common?

2G

18 Which of the following numbers are divisible by 6? a 65 b 121 c 90

d 294

2H

19 Which of the following numbers are divisible by 9? a 162 b 488 c 459

d 49 725

2H

20 Change the last digit to make each of the following numbers divisible by 11. a 654 321 b 764 320 c 111 337 772

2F,G,H

21 State true (T) or false (F) for each of the following. a 146 is divisible by 2. b 3100 is divisible by 5 and 10. c 435 is divisible by 2 and 5. d 144 is divisible by 3. e 7650 is divisible by 8. f 3124 is divisible by 4. g 24 264 is divisible by 3 and 4. h 6045 is divisible by 5 and 8. i 234 is divisible by 6. j 345 098 is divisible by 11. k 240 is divisible by 2, 3, 4, 5, 6 and 8.

2I

22 Simplify: a 112

2I

23 Find: a the 7th square number b the 40th square number.

2I

24 Simplify: 121 a

CHAPTER

test yourself

2

b 92

b

6400

c

c

302

250 000

d 702