© boardworks ltd 2005 1 of 64 n1 integers ks4 mathematics

© Boardworks Ltd 2005 1 of 64

N1 Integers

KS4 Mathematics


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AN1.1 Classifying numbers

N1 Integers

N1.2 Calculating with integers

N1.3 Multiples, factors and primes

N1.4 Prime factor decomposition

N1.5 LCM and HCF


Classifying numbers

Natural numbersPositive whole numbers 0, 1, 2, 3, 4 …

IntegersPositive and negative whole numbers … –3, –2, 1, 0, 1, 2, 3, …

Irrational numbers

Numbers that cannot be expressed in the form n/m, where n and

m are integers. Examples of irrational numbers are and 2.

Rational numbers

Numbers that can be expressed in the form n/m, where n and m

are integers. All fractions and all terminating and recurring decimals are rational numbers, for example, ¾, –0.63, 0.2.

.


Even numbers

Even numbers are numbers that are exactly divisible by 2.

All even numbers end in 0, 2, 4, 6 or 8.

For example, 48 is an even number. It can be written as48 = 2 × 24.

E(3) = 6 E(4) = 8 E(5) = 10

Even numbers can be illustrated using dots or counters arranged as follows:

E(1) = 2 E(2) = 4

The nth even number can be written as E(n) = 2n.


Odd numbers

Odd numbers leave a remainder of 1 when divided by 2.

All odd numbers end in 1, 3, 5, 7 or 9.

For example, 17 is an odd number. It can be written as17 = 2 × 8 + 1

U(1) = 1 U(2) = 3 U(3) = 5 U(4) = 7 U(5) = 9

Odd numbers can be illustrated using dots or counters arranged as follows:

The nth odd number can be written as U(n) = 2n – 1.


Triangular numbers

Triangular numbers are numbers that can be written as the sum of consecutive whole numbers starting with 1.

Triangular numbers can be illustrated using dots or counters arranged in triangles:

For example, 15 is a triangular number. It can be written as15 = 1 + 2 + 3 + 4 + 5

T(1) = 1 T(2) = 3 T(3) = 6 T(4) = 10 T(5) = 15


Triangular numbers

Suppose we want to know the value of T(50), the 50th triangular number.

If we double the number of counters in each triangular arrangement we can make rectangular arrangements:

We could either add together all the numbers from 1 to 50 or we could find a rule for the nth term, T(n).

T(1) = 1 T(2) = 3 T(3) = 6 T(5) = 15

2T(1) = 2 2T(2) = 6 2T(3) = 12

T(4) = 102T(4) = 20 2T(5) = 30


Triangular numbers

Any rectangular arrangement of counters can be written as the product of two whole numbers:

T(1) = 12T(1) = 2

T(2) = 32T(2) = 6

T(3) = 62T(3) = 12

T(4) = 102T(4) = 20

T(5) = 152T(5) = 30

2T(1) = 1 × 2 2T(2) = 2 × 3 2T(3) = 3 × 4 2T(4) = 4 × 5 2T(5) = 5 × 6

From these arrangements we can see that 2T(n) = n(n + 1)So, for any triangular number T(n)

T(n) = n(n + 1)2


Triangular numbers

We can now use this rule to find the value of the 50th triangular number.

T(n) = n(n + 1)2

T(50) = 50(50 + 1)2

T(50) = 50 × 512

T(50) = 25502

T(50) = 1275


Gauss’ method for adding consecutive numbers

There is a story that when the famous mathematician Karl Friedrich Gauss was a young boy at school, his teacher asked the class to add up the numbers from one to a hundred.

The teacher expected this activity to keep the class quiet for some time and so he was amazed when Gauss put up his hand and gave the answer, 5050, almost immediately!


Gauss’ method for adding consecutive numbers

Gauss worked the answer out by noticing that you can quickly add together consecutive numbers by writing the numbers once in order and once in reverse order and adding them together.

For example, to add the numbers from 1 to 10:

1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10

10 + 9 + 8 + 7 + 6 + 5 + 4 + 3 + 2 + 1

11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 + 11 = 110

Sum of the numbers from 1 to 10 = 110 ÷ 2 = 55

Use this method to show that the nth triangular number is:

T(n) = n(n + 1)2


Square numbers

Square numbers are obtained when a whole number is multiplied by itself. They are sometimes called perfect squares.

Square numbers can be illustrated using dots or counters arranged in squares:

For example, 49 is a square number. It can be written as49 = 7 × 7 or 49 = 72.

S(1) = 1 S(2) = 4 S(3) = 9 S(4) = 16 S(5) = 25


Making square numbers

The nth square number S(n) can be written as S(n) = n2.

We can multiply a whole number by itself. For example, 25 = 5 × 5 or 25 = 52.

We can add consecutive odd numbers starting from 1. For example, 25 = 1 + 3 + 5 + 7 + 9.

We can add together two consecutive triangular numbers. For example, 25 = 10 + 15

There are several ways to generate a sequence of square numbers.

We can find the product of two consecutive even or odd numbers and add 1. For example, 25 = 4 × 6 + 1.


Difference between consecutive squares

Show that the difference between two consecutive square numbers is always an odd number.

If we use the general form for a square number n2, where n is a whole number, we can write the square number following it as

The difference between two consecutive square numbers can therefore be written as

(n + 1)2 – n2 = (n + 1)(n + 1) – n2

= n2 + n + n + 1 – n2

= 2n + 1

2n + 1 is always an odd number for any whole number n.

(n + 1)2.


Cube numbers

Cube numbers are obtained when a whole number is multiplied by itself and then by itself again.

Cube numbers can be illustrated using spheres arranged in cubes:

For example, 64 is a cube number. It can be written as64 = 4 × 4 × 4 or 64 = 43.

C(2) = 8 C(3) = 27 C(4) = 64 C(5) = 125C(1) = 1

The nth cube number C(n) can be written as C(n) = n3.


Squares, triangles and primes


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N1 Integers



N1.5 LCM and HCF

N1.1 Classifying numbers


Negative numbers

A positive or negative whole number, including zero, is called an integer.

For example, –3 is an integer.

–3 is read as ‘negative three’.

This can also be written as –3.

It is 3 less than 0.

0 – 3 = –3Here the ‘–’ sign means minus 3 or subtract 3.

We say, ‘zero minus three equals negative three’.

Here the ‘–’ sign means negative 3.


Positive and negative integers can be shown on a number line.

Positive integersNegative integers

We can use the number line to compare integers.

For example,

–3–8

–3 > –8

–3 ‘is greater than’ –8

Integers on a number line


Adding integers

We can use a number line to help us add positive and negative integers.

–2 + 5 =

-2 3

= 3

To add a positive integer we move forwards up the number line.


We can use a number line to help us add positive and negative integers.

To add a negative integer we move backwards down the number line.

–3 + –4 == –7

-3-7

–3 + –4 is the same as –3 – 4

Adding integers


5-3

Subtracting integers

We can use a number line to help us subtract positive and negative integers.

5 – 8 == –3

To subtract a positive integer we move backwards down the number line.

5 – 8 is the same as 5 – +8


3 – –6 =

3 9

= 9


To subtract a negative integer we move forwards up the number line.

3 – –6 is the same as 3 + 6




–4 – –7 =

-4 3

= 3


–4 – –7 is the same as –4 + 7



Adding and subtracting integers

To add a positive integer we move forwards up the number line.

To add a negative integer we move backwards down the number line.

To subtract a positive integer we move backwards down the number line.


a + –b is the same as a – b.

a – –b is the same as a + b.


Integer circle sums


When multiplying negative numbers remember:

Rules for multiplying and dividing

Dividing is the inverse operation to multiplying.

When we are dividing negative numbers similar rules apply:

+ × + = +

–+ × = –

–+× =–

– +× =–

+ ÷ + = +

–+ ÷ = –

–+÷ =–

– +÷ =–


Multiplying and dividing integers

Complete the following:

–3 × 8 =

42 ÷ = –6

× –8 = 96

47 × = –141

–72 ÷ –6 =

–36 ÷ = –4

÷ –90 = –6

–7 × = 175

–4 × –5 × –8 =

3 × –8 ÷ = 1.5

–24

–7

–12

–3

12

9

540

–25

–160

–16


Using a calculator

We can enter negative numbers into a calculator by using thesign change key: (–)

For example:

–456 ÷ –6 can be entered as:

(–) 4 5 6 ÷ (–) 6 =

The answer will be displayed as 76.

Always make sure that answers given by a calculator are sensible.


What two integers have a sum of 2 and a product of –8?

Sums and products

Start by writing down all of the pairs of numbers that multiply together to make –8.

Since –8 is negative, one of the numbers must be positive and one of the numbers must be negative.

We can have:

–1 × 8 = –8 1 × –8 = –8 –2 × 4 = –8 or 2 × –4 = –8

–1 + 8 = 7 1 + –8 = –7 –2 + 4 = 2 2 + –4 = –2

The two integers are –2 and 4.


Sums and products


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N1 Integers


N1.5 LCM and HCF




Multiples

A multiple of a number is found by multiplying the number by any whole number.

What are the first six multiples of 7?

To find the first six multiples of 7 multiply 7 by 1, 2, 3, 4, 5 and 6 in turn to get:

7, 14, 21, 28, 35 and 42.

Any given number has infinitely many multiples.


Factors

A factor (or divisor) of a number is a whole number that divides into it exactly.

Factors come in pairs. For example,

What are the factors of 30?

1 and 30, 2 and 15, 3 and 10, 5 and 6.

So, in order, the factors of 30 are:

1, 2, 3, 5, 6, 10, 15 and 30.


Prime numbers

If a whole number has two, and only two, factors it is called a prime number.

For example, the number 17 has only two factors, 1 and 17.

Therefore, 17 is a prime number.

The number 1 has only one factor, 1.

Therefore, 1 is not a prime number.

There is only one even prime number. What is it?

2 is the only even prime number.


Prime numbers

There are 25 prime numbers less than 100.

These are:2

13

31

53

73

3

17

37

59

79

5

19

41

61

83

7

23

43

67

89

11

29

47

71

97

What if we go above 100? Around 400 BC the Greek mathematician, Euclid, proved that there are infinitely many prime numbers.


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N1 Integers

N1.5 LCM and HCF





A prime factor is a factor that is a prime number.

For example,

What are the prime factors of 70?

The factors of 70 are:

1 2 5 7 10 14 35 70

The prime factors of 70 are 2, 5, and 7.

Prime factors


= 2 × 5 × 770

2 × 2 × 2 × 756 = This can be written as 56 = 23 × 7

= 3 × 3 × 1199 This can be written as 99 = 32 × 11

Every whole number greater than 1 is either a prime number or can be written as a product of

two or more prime numbers.

Products of prime factors


The prime factor decomposition

When we write a number as a product of prime factors it is called the prime factor decomposition or prime factor form.

For example,

The prime factor decomposition of 100 is:

There are two methods of finding the prime factor decomposition of a number.

100 = 2 × 2 × 5 × 5

= 22 × 52


36

4 9

2 2 3 3

36 = 2 × 2 × 3 × 3

= 22 × 32

Factor trees


36

3 12

4 3

2 2

36 = 2 × 2 × 3 × 3

= 22 × 32

Factor trees


2100

30 70

6 5

2 3

10 7

2 5

2100 = 2 × 2 × 3 × 5 × 5 × 7

= 22 × 3 × 52 × 7

Factor trees


780

78 10

392

3 13

25

780 = 2 × 2 × 3 × 5 × 13

= 22 × 3 × 5 × 13

Factor trees


962

482

242

122

62

33

1

2

2

2

2

2

3

96 = 2 × 2 × 2 × 2 × 2 × 3

= 25 × 3

Dividing by prime numbers


3153

1053

355

77

1

3

3

5

7

315 = 3 × 3 × 5 × 7

= 32 × 5 × 7



7022

3513

1173

393

1313

1

2

3

3

3

13

702 = 2 × 3 × 3 × 3 × 13

= 2 × 33 × 13



Using the prime factor decomposition

Use the prime factor form of 324 to show that it is a square number.

3242

1622

813

273

93

33

1

2

2

3

3

3

3

324 = 2 × 2 × 3 × 3 × 3 × 3

= 22 × 34

This can be written as:

(2 × 32) × (2 × 32)

or (2 × 32)2

If all the indices in the prime factor decomposition of a number are even, then the number is a square number.

If all the indices in the prime factor decomposition of a number are even, then the number is a square number.



Use the prime factor form of 3375 to show that it is a cube number.

33753

11253

3753

1255

255

55

1

3

3

3

5

5

5

3375 = 3 × 3 × 3 × 5 × 5 × 5

= 33 × 53

This can be written as:

(3 × 5) × (3 × 5) × (3 × 5)

or (3 × 5)3

If all the indices in the prime factor decomposition of a number are

multiples of 3, then the number is a cube number.

If all the indices in the prime factor decomposition of a number are

multiples of 3, then the number is a cube number.



168 = 23 × 3 × 7 4116 = 22 × 3 × 73 294 = 2 × 3 × 72

Use the prime factor decompositions of the numbers given above to answer the following questions.

1) What is 168 × 294 as a product of prime factors?

168 × 294 = (23 × 3 × 7) × (2 × 3 × 72 )

= 23 × 2 × 3 × 3 × 7 × 72

= 24 × 32 × 73



168 = 23 × 3 × 7 4116 = 22 × 3 × 73 294 = 2 × 3 × 72


2) What is 4116 ÷ 294?

4116 ÷ 294 =22 × 3 × 73

2 × 3 × 72

=2 × 2 × 3 × 7 × 7 × 7

2 × 3 × 7 × 7

= 2 × 7

= 14



168 = 23 × 3 × 7 4116 = 22 × 3 × 73 294 = 2 × 3 × 72


3) Is 4116 divisible by 168?

If we divide 4116 by 168 we have:

4116 ÷ 168 =22 × 3 × 73

23 × 3 × 7

=2 × 2 × 3 × 7 × 7 × 7

2 × 2 × 2 × 3 × 7

There is a 2 left in the denominator

No, 4116 is not divisible by 168.



168 = 23 × 3 × 7 4116 = 22 × 3 × 73 294 = 2 × 3 × 72


4) Show that 294 × 6 is a square number.

We can write 6 as 2 × 3

294 × 6 = 2 × 3 × 72 × 2 × 3

Rearranging,

294 × 6 = 2 × 2 × 3 × 3 × 72

= 22 × 32 × 72

= (2 × 3 × 7)2



168 = 23 × 3 × 7 4116 = 22 × 3 × 73 294 = 2 × 3 × 72


5) Write the fraction in its simplest form.168294

168294

= 23 × 3 × 72 × 3 × 72

2 × 2 × 2 × 3 × 72 × 3 × 7 × 7

=

= 47


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N1.5 LCM and HCF

N1 Integers






The lowest common multiple

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

For small numbers we can find this by writing down the first few multiples for both numbers until we find a number that is in both lists.

For example,

Multiples of 20 are : 20, 40, 60, 80, 100, 120, . . .

Multiples of 25 are : 25, 50, 75, 100, 125, . . .

The LCM of 20 and 25 is 100.


We use the lowest common multiple when adding and subtracting fractions.

For example,

Add together 4

9

5

12and

The LCM of 9 and 12 is 36.

+4

9

5

12=

36

× 4

× 4

16+

36

× 3

× 3

15=

31

36

The lowest common multiple


The highest common factor

The highest common factor (or HCF) of two numbers is the highest number that is a factor of both numbers.

We can find the highest common factor of two numbers by writing down all their factors and finding the largest factor in both lists.

For example,

Factors of 36 are :

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

Factors of 45 are :

1, 3, 5, 9, 15, 45.

The HCF of 36 and 45 is 9.


We use the highest common factor when cancelling fractions.

For example,

Cancel the fraction 36

48

The HCF of 36 and 48 is 12, so we need to divide the numerator and the denominator by 12.

36

48=

÷12

3

÷12

4

The highest common factor


Using prime factors to find the HCF and LCM

We can use the prime factor decomposition to find the HCF and LCM of larger numbers.

For example,

Find the HCF and the LCM of 60 and 125.

602302153551

60 = 2 × 2 × 3 × 5

29421473497771

294 = 2 × 3 × 7 × 7


60 294

60 = 2 × 2 × 3 × 5

294 = 2 × 3 × 7 × 7

22

35

7

7

HCF of 60 and 294 = 2 × 3 = 6

LCM of 60 and 294 = 2 × 5 × 2 × 3 × 7 × 7 = 2940



The LCM of co-prime numbers

If two numbers have a highest common factor (or HCF) of 1 then they are called co-prime or relatively prime numbers.

For two whole numbers a and b we can write:

If two whole numbers a and b are co-prime then:

For example, the numbers 8 and 9 do not share any common multiples other than 1. They are co-prime.

Therefore, LCM(8, 9) = 8 × 9 = 72

a and b are co-prime if HCF(a, b) = 1a and b are co-prime if HCF(a, b) = 1

LCM(a, b) = abLCM(a, b) = ab


The LCM of numbers that are not co-prime

If two numbers are not co-prime then their highest common factor is greater than 1.

If two numbers a and b are not co-prime then their lowest common multiple is equal to the product of the two numbers divided by their highest common factor.

We can write this as:

For example,

LCM(a, b) = ab

HCF(a, b)

LCM(8, 12) =8 × 12

HCF(8, 12)=

96

4= 24

© boardworks ltd 2005 1 of 64 n1 integers ks4 mathematics

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