Download Higher Practice Book Unit 2 Chapter 1

Unit 2
1.1 Understanding prime factors, LCM and HCF
1 Number
Key Points
square root (√): the opposite
of the square of a
_
number. E.g. If 3  3  9, √9  3.
cube number: a number that is the result of
cubing a whole number.
cube root (³√): the opposite of the
cube of a
__
3
number. E.g. If 3  3  3  27, √27  3.
BIDMAS: the order of number operations.
Brackets, Indices, Divide, Multiply, Add,
Subtract.
index number: a number written in the form an.
laws of indices:
am  an  amn
am  amn
___
an
(am)n  amn
finding the square of a number: multiply the
number by itself.
finding the cube of a number: multiply the
number by itself and then multiply the result by the
original number.
factors of a number: whole numbers that divide
exactly into the number. They always include 1 and
the number itself.
multiples of a number: the results of
multiplying the number by positive whole numbers.
common multiple: a number that is a multiple
of two or more numbers.
prime number: a whole number greater than 1
whose only factors are itself and 1.
prime factor: a factor that is also a prime
number. Any number can be written as a product
of its prime factors.
common factor: a number that is a factor of two
or more numbers.
lowest common multiple (LCM): the lowest
multiple that is common to two or more numbers.
highest common factor (HCF): the highest
factor common to two or more numbers.
square number: a number that is the result of
squaring a whole number.
1.1 Understanding prime factors,
LCM and HCF
C
5
a Write 24 and 56 as products of their prime
factors.
b Find the HCF of 24 and 56.
c Find the LCM of 24 and 56.
6
a Write 36 and 60 as products of their prime
factors.
b Find the HCF of 36 and 60.
c Find the LCM of 36 and 60.
7
Find the HCF and LCM of the following pairs of
numbers.
a 28 and 70
b 64 and 84
c 72 and 108
d 132 and 168
8
c  22  33  5 , d  23  32  7
a Find the HCF of c and d.
b Find the LCM of c and d.
9
s  23  33  5  7, t  22  53
a Find the HCF of s and t.
b Find the LCM of s and t.
Exercise 1A
Questions in this chapter are targeted at the grades indicated.
C
1
A03
Can the difference between two prime numbers
be a prime number?
Explain your answer.
[Hint: Try testing pairs of prime numbers.]
2
The number 160 can be written in the form
2n  5.
Find the value of n.
3
The number 132 can be written in the form
2p  q  r where p, q and r are prime numbers.
Find the values of p, q and r.
4
Find the HCF and LCM of the following pairs of
numbers.
a 4 and 6
b 10 and 15
c 4 and 12
d 12 and 18
37
38
Chapter 1 Number
B 10
11
A03
12
A03
A 13
A03
Unit 2
Bertrand’s theorem states that ‘Between any
two numbers n and 2n, there always lies at least
one prime number, providing n is bigger than 1’.
Show that Bertrand’s theorem is true:
a for n  5
b for n  12
c for n  25.
Frank has two flashing lamps. The first lamp
flashes every 4 seconds. The second lamp
flashes every 6 seconds. Both lamps start
flashing together.
a After how many seconds will they again flash
together?
b How many times in a minute will they flash
together?
A shop is going to order matching jackets and
trousers. The jackets come in boxes of 5 and the
trousers come in boxes of 6.
What is the smallest number of boxes of jackets
and trousers that the shop must order to ensure
that there is a jacket for every pair of trousers?
Exercise 1C
1
2
Work out
a 33
e 142
b 53
Work__ out
a √9
__
___
b √ 49
D
C
3
4
c √ 121
Work out
a (8)2
b (3)3
c (7)2
d (2)3
e (10)2
__
Work__out
3
a √1
___
3
c √8
__
b √ 49  52
c 53  √ 9
3
d √27  43
____
__
____
__
f 103  53
2
h 43  (7)
__
√ 36
__
j 23  ____
3
√8
____
3
√ 64
122 __
___________
l
√ 16
___
√ 225
__
k 10  _____
√ 25
3
Exercise 1B
D
1
2
Write down:
a the first
12 square
numbers
b the first 6
cube
numbers.
1.3 Understanding the order of operations
Examiner’s Tip
You need to be able to recall
• integer squares from
2  2 up to 15  15
and the corresponding
square roots
• the cubes of 2, 3, 4, 5 and
10.
From each
list write down all the numbers which are:
i square numbers
ii cube numbers.
a
b
c
d
______
3
e √1000
Work out
a 33  22
3
e √1000  √ 25
g (1)3  43  (5)2
92
i __
33
1.2 Understanding squares and cubes
e √ 100
3
d √27
__
Ben says that, if you find the average of two
prime numbers, you will always get a whole
number. Is Ben correct?
Give a reason for your answer.
___
d √ 16
Remember, when multiplying or
dividing:
two signs the same give a 
two different signs give a 
_____
5
d 102
Examiner’s Tip
3
b √125
B
c 42
50, 20, 40, 30, 4, 80, 27, 36, 25
64, 21, 9, 57, 60, 10, 7, 100, 48, 35, 90, 1
123, 25, 75, 105, 50, 125, 48, 81, 169
100, 175, 125, 93, 64, 75, 8, 200, 1000
Exercise 1D
D
1
2
Work out
a 5  (1  4)
c 35  5  2
e (7  3)  2
g 24  (7  3)
i 7  (5  1)
k 5343
m 14  2  9
o 3542
b
d
f
h
j
l
n
p
Work out
a (4  5)2
c 2  (7  2)2
e 2  (5  1)2
(17  3)2
g ________
22  3
b 12  22
d 2  42__
 2  52
f 2  √36  2  23
32  22
h ______
5
363
24  (5  1)
842
56  8  2
946
32  8  5
7  2  4  12  3
(21  9)  (3  4)
Unit 2
B
3
1.4 Understanding the index laws
C
Work out
___
3
a (4  1)3  √125
b ((12  2)  4)  ((1  4)  2)
___
c 92  33  √225  1  5
______
__
3
3
d (√64  3)2  √23  53
3
a 7n  72  75
c 3n  35  315
e 66  65  6n
4
1.4 Understanding the index laws
Exercise 1E
C
1
5
Write as a power of a single number.
a 54  57
b 27  23
c (92)4
d 67  64
2
B
e 49  43
b 46  43
e 22  2
6
c (32)2
b 56  5n  52
d 9n  97  911
Write as a power of a single number.
66  68
87  84
45  48
a ______
b ______
c ______
2
4
8
6
46
13
5
5
5
2 2
d ______
e ______
53  55
2  24
Work out
56
112  115
45  42
a ________
b ______
c ______
6
4
3
11
4
5 5
5
5
9
3
10

10

3
______
________
d
e
37
104  103
Work out the value of n in the following.
a 20  5  2n
d 24  3  2n
Work out
a 105  103
d 55  52
Find the value of n
b 64  2n c 135  3n  5
e 162  2  3n
39