CHAPTER TWO

Chapter 2: WORKING WITH INTEGERS

Learning objectives

By the end of this topic, the learners should be able to

• Identify ,read and write natural numbers as numerals and words in million, billion and trillion

• Differentiate between natural numbers and whole numbers /integers

• Identify directed numbers

• Use directed numbers (limited to integers) in real life situations

• Use the hierarchy of operations to carry out the four mathematical operations on integers

• Identify Even, Odd, Prime and Composite Numbers

• Find the prime factors of any number

• Relate common factors with HCF and multiples with LCM

• Work out and use of divisibility tests of some numbers

2.1 Natural numbers

Introduction

Natural numbers can be classified into various groups of numbers. In your primary education, you learnt numbers such as even, odd, prime and composite.

Activity: Writing and reading numbers

There are two boxes. In one box, number cards are written in gures and the others in words.

In groups, a member picks one card from one of the boxes. After all the cards have been picked, one member displays his/her card; then the others check their cards, and the matching card is displayed.

EXAMPLES

1. Write 999; 444; 230; 999 in words.

Nine hundred ninety nine billion,four hundred forty four million Two hundred thirty thousand nine hundred ninety nine.

2. Write 940; 340; 400; 230; 886 in words.

Nine hundred forty trillion three hundred forty billion, four hundred million Two hundred thirty thousand eight hundred eighty six.

3. Write: Nine hundred ninety nine trillion seven hundred eighty eight billion, five hundred

ninety nine million nine hundred ninety nine thousand eight hundred eighty six in figures.

2.1 Exercise Set

1. Write the following in words:

(a) 6,800

(b) 9,888,008

(c) 722,820,060

(d) 76000

(e) 8,888,888

(f) 9,770,500

(g) 8,999,909,700

(h) 6,745,842,003

(i) 3,730,284,654,040

2. Write the following in gures

(a) Seven hundred three million seven thousand and six

(b) Four billion seventy-nine million ve thousand six

(c) One trillion three hundred forty billion seven hundred seventy-ve million two hundred sixty thousand

(d) Nine hundred ninety- nine trillion seven hundred eighty eight billion five hundred ninety nine million nine hundred ninety nine thousand eight hundred eighty six

(e) Seventy seven million two hundred sixty seven thousand nine hundred eighty

2.1.1 Differentiating between natural numbers and whole numbers/integers

Activity: Relating natural numbers and integers

Two learners Hannah and Ritah went to the school canteen to buy some snacks for their breakfast.

Ritah bought 3 pancakes at UGX.200 each and 1 ban at UGX. 300. Hannah checked her bag and found out that her money was stolen. She borrowed some money from Ritah. She bought four 4 pancakes and 2 bans.

Questions

1. Which of the two learners had more money?

2. How much money did Hannah borrow from Ritah?

3. Ritah said that Hannah had negative UGX. 1400. Was she correct?

4. Give reasons for your answer.

2.2 Use Directed Numbers (Limited to Integers) in Real-life Situations

Numbers which have a direction and a size are called directed numbers. Once a direction is chosen

as positive (+), the opposite direction is taken as negative (-)

Activity: Integers in real-life situations

Read the story below and answer the questions.

Once upon a time, there lived an old woman. She had hot and cold stones and a big pot of water.

If she put one hot stone in the water, the temperature of the water would rise by 1 degree. If she took the hot stone out of the water again, the temperature would go down by 1 degree .i.e if the temperature of the water was 28 degrees and the old woman removes 2 hot stones , the temperature would drop to 26 degrees ,and if the temperature of the water was at 85 degrees and the old woman adds 4 hot stones, the temperature would rise to 89 degrees.

Questions

1. If the temperature of the water is 24 degrees and the old woman adds 5 hot stones, what is the new temperature of the water?

2. Now imagine that the temperature of the water is at 29 degrees. The old woman takes a spoon and takes out 3 of the hot stones from the pot. What is the temperature of the water when the old woman removes 3 hot stones? Explain your answer.

3. The old woman also had cold stones. If she adds 1 cold stone to the water, the temperature goes down by 1 degree. The temperature of the water was 26 degrees. Then the old woman added 4 cold stones. What is the temperature of the water after the old woman added 4 cold stones

4. Give a reason for your answer.

5. Imagine that the temperature of the water was 22 degrees and the old woman removes 3 cold stones. What happens to the temperature of the water?

6. What is the new temperature of the water? Explain your answer.

2.3 Use the Hierarchy of Operations to Carry out the Four Mathematical Operations on Integers

Activity: Operation on integers

1. Sarah moved 5 steps to the right from a fixed point. Then she moved 9 steps to the left.

(a) How far is Sarah from the fixed point?

(b) Peter gave his answer as 4 steps to the left of the fixed point and John as 4 (negative)

4). Who is correct? Give reasons for your answer.

2. A group of learners of Geography went for a tour to Kabale. They found out that the temperature at one time was 130C. At around mid-night the temperature was 100C. By how many degrees had the temperature dropped?

2.4 Number line

Integers can be illustrated on a number line (number scale) as shown below

Positive integers are to the right of a zero and negative integers are to the left of zero. Positive integers are shifts to the right while negative integers are shifts to the left

2.4.1 Addition of numbers on a number line

Work out -4 ++ 6 using a number line

2.4.2 Subtraction of numbers on a number line

Work out -3 - 4 using a number line

2.4.3 Multiplication of numbers on a number line

Multiplication is interpreted as repeated addition of positive or negative numbers.

Work out +2 ×+ 3

SOLUTION

+2 ×+ 3 =+ 2 ++ 2 ++ 2

2.4.4 Division of numbers on a number line

Division is interpreted as repeated subtraction of positive or negative numbers

STEPS

• Draw an empty number line.

• Start from the right hand side of the number line .i.e From the Dividend

• Subtract by groups i.e subtract the divisor from the dividend up to when you reach zero.

• Count the jumps made from the dividend

(i) Workout +6 ÷+ 2 using a numberline

3 is the number of times you can subtract 2 from 6 before you get to zero i.e 3 represents the jumps made (skip 2 digits from 6)

5 is the number of times you can subtract 3 from 15 before you get to zero i.e 5 represents the jumps made (skip 3 digits from 15)

2.2 Exercise Set

1. Work out the following in degrees

2. Workout the following numbers using a number line

(a) +3 ++ 5

(b) +4 - 6

(c) -7 -- 3

(d) +4 -+ 8

(e) +5 -- 3

(f) 4 ×+ 2

(g) +3 ×- 3

(h) +4 ×- 2

(i) +23 ×+ 6

(j) +3 -- 4

(k) +9 ÷ 3

(l) 24 ÷ 4

3. Work out the following:

(a) +32 +- 5

(b) +84 - 6

(c) -17 -- 13

(d) +104 -+ 5

(e) +51 -- 32

(f) 42 ×+ 2

(g) +13 ×- 3

(h) +74 ×- 2

(i) +123 ×+ 6

(j) +73 -- 4

(k) +99 ÷ 3

(l) 124 ÷ 4

4. Work out

Hint:BODMAS MUST BE APPLIED

(a) +3 ×- 4 ×- 6

(b) +4 ×- 2 ×+ 5

(c) +7 ×- 8 ×+ 4

(d) -20 ×- 6 ÷+ 2

(e) -25 ÷ 5 ×- 8

(f) -34 ×+ 2 ÷+ 2

(g) 24 of 13˘(18 ÷ 6 + 3) ÷ (9 × 3 - 25)

(h) 89 - (99 - 84 ÷ 2 + 2)

(i) 6 ÷ (2 + (2 × 6 - 2))

(j) 4 of (4 + 3) - 2(1 + 9) ÷ 4

5. In a certain mathematics test a correct answer scores 5 marks and an incorrect answer, the child gets a penalty of two marks deducted. Joy guessed all the answers. She got 12 correct and 8 wrong. Work out her total marks.

2.5 Identify Even, Odd, Prime and Composite Numbers

Introduction

Natural numbers can be classified into various groups of numbers. In your primary education, you learnt numbers such as even, odd, prime and composite.

Activity: Identifying even, odd, prime and composite numbers

1. Natural Numbers

These are numbers used in counting.e.g N={1; 2; 3; 4 · · · }

2. Whole Numbers

These are counting numbers including zero.e.g W={0; 1; 2; 3; 4 · · · }

3. Square Numbers

These are numbers got after multiplying a natural number by itself e.g S={1; 4; 9; 16 · · · }

4. Cube Numbers

These are numbers got after multiplying a natural number three times e.g C={1; 8; 27; 64 · · · }

5. Even Numbers

This is a number that is exactly divisible by two .e.g E={2; 4; 6; 8 · · · }

6. Prime Numbers

This is a number with only two factors one and itself .e.g E={2; 3; 5; 7 · · · }

7. Composite Numbers

This is a number with more than two factors .e.g {4; 6; 8 · · · }

EXAMPLES

1. Identifying prime and composite numbers

2. Are there numbers that belong to more than one group?

3. What is the 6th prime number?

4. What is the 19th composite number?

2.3 Exercise Set

1. (a) The table below shows the natural numbers from 1 to 100.Color the numbers

(b) What is the 7th odd number?

(c) What is the 20th even number?

(d) Are there numbers that belong to more than one group?

2. Write down a number which is both an even number and a prime number

3. List the rst 10 composite numbers

2.6 Finding the Prime Factors and multiples of any Number

• Factors

Factors of numbers are all numbers that divide exactly into it, leaving no remainder..e.g 6 is

divisible by {1; 2; 3; 6} Therefore the factors of 6 are F6={1; 2; 3; 6}

• Prime Factor

This is a factor which is a prime number.

• Prime Factorisation

Expresses a number as a product of only its prime factors .

• Multiple of a Number

Is that number multiplied by another integer .i.e When two numbers are multiplied together, the product is called multiple .e.g Multiples of 5 include {5; 10; 15; 20 · · · }.

EXAMPLES

1. List all the factors of the following numbers

(a) 12

F12 = {1; 2; 3; 4; 6; 12}

(b) 32

F32 = {1; 2; 4; 8; 16; 32}

(c) 60

F60 = {1; 2; 3; 4; 5; 10; 12; 15; 30; 60}

2. List the multiples of the following numbers

(a) 2

M2 = {2; 4; 6; 8; 10; · · · }

(b) 3

M3 = {3; 6; 9; 12; · · · }

(c) 12

M12 = {12; 24; 36; 48; 60; · · · }

3. Express each of the following numbers as a product of its prime factors

(a) 36

2.7 Work Out and Use Divisibility Tests of Some Numbers

• If a number is divisible by 2, its last digit is even(2,4,6,8) or 0.

• If a number is divisible by 3, the sum of its digits will be a multiple of 3.

2.7. WORK OUT AND USE DIVISIBILITY TESTS OF SOME NUMBERS

• If a number is divisible by 4, the last two digits will be a multiple of 4.

• If a number is divisible by 5, it will end in 0 or 5.

• If a number is divisible by 6, its last digit is even and the sum of its digits is divisible by 3

• If a number is divisible by 8, its lastthree digits form a number divisible by 8 .

• If a number is divisible by 9, the sum of its digits will be a multiple of 9.

• If a number is divisible by 10, its last digit is 0

2.4 Exercise Set

1. List all the common divisors/ factors of the following:

(a) 16

(b) 60

(c) 112

(d) 225

(e) 90

(f) 100

(g) 18

(h) 48

2. List down multiples of the following numbers that are less than 50

(a) 5

(b) 20

(c) 9

(d) 10

(e) 13

(f) 7

(g) 24

(h) 11

3. Find the prime factors of the following numbers. Give your answer in power form(Power notation).

(a) 28

(b) 54

(c) 204

(d) 156

(e) 225

(f) 1020

(g) 132

(h) 90

(i) 1232

(j) 993

(k) 2145

(l) 780

4. (a) List all the factors of each of the following numbers: 11, 12, 13, 14, 15, 16, 17, 18, 19,

20

(b) Which of these numbers are prime?

5. Explain why 99 is not a prime number.

6. Which of the following are prime numbers: 33, 35, 37, 39 ?

7. Find the prime factors of 72.

8. (a) Find the prime factors of 40.

(b) Find the prime factors of 70.

(c) Which prime factors do 40 and 70 have in common?

9. Find the prime factors that 48 and 54 have in common.

10. A number has prime factors 2, 5 and 7. Which is the smallest number that has these prime factors?

2.8. RELATE COMMON FACTORS WITH HCF AND MULTIPLES WITH LCM

11. The first 5 prime numbers are 2, 3, 5, 7 and 11. Which is the smallest number that has these prime factors?

12. Write down the rst two prime numbers which are greater than 100.

13. Which is the rst prime number that is greater than 200?

14. Use a factor tree to nd the prime factors of:

(a) 102

(b) 60

(c) 30

(d) 80

(e) 200

(f) 72

15. A number is expressed as the product of its prime factors as 52 × 72.What is the number?

16. A number is expressed as the product of its prime factors as 23 × 35.What is the number?

17. The prime factors of a number are 2, 7 and 11. Which are the three smallest numbers with these prime factors?

18. Given the following numbers: 12, 132, 1212, 3243, 1112, 81, 18, 27, 279, 2580, 5750. Find out which of them are divisible by:

(a) 2

(b) 3

(c) 4

(d) 5

(e) 6

(f) 7

(g) 8

(h) 9

(i) 10

2.8 Relate Common Factors with HCF and Multiples with LCM

In this section we deal with finding LCM and HCF. We use the knowledge of multiples and factors.

2.8.1 Highest Common Factor(HCF)

• Highest Common Factor(H.C.F)

Is the highest number that divides exactly in two or more numbers .H.C.F is also called Greatest Common Divisor(G.C.D) or The highest common factor (HCF) of two numbers is the largest number that is a factor of both.

STEPS

To find the HCF of two or more numbers:

• Express each of the numbers as a product of prime factors,

• Pick out the least power of each common factor. The product of these gives the HCF or GCF

EXAMPLES

2.8. RELATE COMMON FACTORS WITH HCF AND MULTIPLES WITH LCM

1. Find the HCF of 12 and 15.

2.5 Exercise Set

1. (a) Write the factors of 8 and 12

(b) Identify the common factors of 8 and 12

(c) What is the highest common factor

30 Practice makes mathematics easier

2.8. RELATE COMMON FACTORS WITH HCF AND MULTIPLES WITH LCM

2. Find the HCF of the following:

(a) 96, 57

(b) 49 ,84

(c) 72,144,288

(d) 42 ,63 ,105

(e) 28,42,98

(f) 132, 156,204,228

(g) 54, 48

(h) 42 ,63 ,105

(i) 90,126,270

3. Find the HCF of:

(a) 6 and 9

(b) 14 and 18

(c) 30 and 24

(d) 15 and 10

(e) 90 and 120

(f) 96 and 72

(g) 56 and 60

(h) 77 and 50

(i) 300 and 550

(j) 320 and 128

(k) 46 and 62

(l) 124 and 72

4. (a) Use a factor tree to nd the prime factorisation of 42.

(b) Use a factor tree to nd the prime factorisation of 90.

(c) Find the HCF of 42 and 90.

5. Stephen has two pieces of cloth. One piece is 36 inches wide and the other piece is 24 inches wide. He wants to cut both pieces into strips of equal width that are as wide as possible.

How wide should he cut the strips?

6. Determine the smallest sum of money out of which a number of men, women and children may receive UGX. 750, Ush.900 and Ush.700 each.

2.8.2 Lowest Common Multiple(LCM)

Lowest common multiple(L.C.M)

The lowest common multiple (LCM) of two numbers is the smallest number that is a multiple of both.

EXAMPLES

1. What is the LCM of 5 and 7

M5 = {5; 10; 15; 20; 25; 30; 35 ; 40 · · · }

M7 = {7; 14; 21; 28; 35 ; 42 · · · }

The LCM of 5 and 7 is 35.

2.8. RELATE COMMON FACTORS WITH HCF AND MULTIPLES WITH LCM

2. Find the LCM of 16, 12 and 24.