Download Page 1 of 11 Secondary 1 Mathematics Topic 1 – Prime Numbers

Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Learning Objectives
1. Identify prime numbers of up to 150
2. Express composite numbers as products of their prime factors.
3. Find highest common factors (HCF) and lowest common multiples (LCM) of a set of a
composite numbers.
4. Solve word problems using HCF and LCM concepts.
Section 1: Prime Numbers
Introduction
Prime numbers are whole numbers greater than 1, that have only two factors: 1, and the
number itself. Here’s a quick exercise to help you find all prime numbers up to 100.
The Prime Sieve
Look at the table on the right. The
following steps will help you to find
all the prime numbers.
Steps
1. Cancel out the number 1
2. Cancel out all multiples of 2
1. 3. Cancel out all multiples of 3
2. 4. Cancel out all multiples of 5
3. 5. Cancel out all multiples of 7
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
32
34
35
36
37
38
39
40
41
42
42
44
45
46
47
48
49
50
51
52
52
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
90
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Once you have done that, highlight the remaining numbers. All these remaining numbers are
prime numbers. You see that they are not multiples of any numbers except 1 and
themselves.
Numbers that have more than two factors are called composite numbers (e.g. 4, 9. 16 etc..)
Note that 1 is neither a composite nor a prime number.
Why is this important?
Prime numbers are very important in the real world. They are used in many places especially
an area of technology called encryption and cryptography. Encryption is an area of science
which uses prime numbers to mix up information and data into a code, so that no one but the
intended recipient with the prime number key knows how to decode it. This is important in
our everyday life especially in internet security, from logging on to Facebook to sending an
email, or paying for things with credit cards and even making a phone call. Encryption
ensures that all our private information is secure and not accessible by strangers.
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Page 1 of 11
Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Exercise 1: Prime Numbers
1. List all prime numbers between 0 and 20.
2. The number 16 can be expressed as a sum of two prime numbers. Give two such
expressions.
3. a) Write down all prime numbers between 1 and 100 that end in 7.
b) Write down all prime numbers between 1 and 100 that end in 9.
4. Find the difference between the largest and smallest prime number between 30 and 70.
5. List all the prime numbers in the following series:
3
144
3
a) 13 , 5 , , 67 , √64 , 27 , 43 ,
5
12
121
b) √9 , 1 , 17 , 56 , 3.14 , 67 ,
and 51
11
6. Find the sum of all prime numbers between 40 and 55.
7. Complete the following table:
Pair of prime numbers
Product
Is the product even or odd?
2 and 3
2×3=6
Even
2 and 5
3 and 5
3 and 7
5 and 7
2 and 13
a. When two prime numbers are odd, what can you say about their product?
b. When one of the prime numbers in the pair is even, what can you say about their
product?
c. Can the product of two prime numbers be a prime number?
8.
If 𝑝 and 𝑞 are prime numbers between 10 and 15, state whether the following are true or
false:
a. 𝑝𝑞 − 1 is a composite number.
b. 𝑞2 is a composite number.
c. 𝑝2 𝑞 only has four factors.
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Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Section 2: Prime Factorisation
Recall what composite numbers are. They are numbers that have more than two factors.
Prime factorisation is to express any composite number as a product of their prime factors
only.
Example – Express the number 72 in terms of its prime factors.
1. Factor Tree
You can find all the prime factors of 72 using a factor tree:
a. Write down the number 72
b. Find a prime factor of 72. (e.g. 2)
c. Write the prime factor on the left.
d. Write the quotient (72 ÷ 2) under 72.
e. Repeat the process for the quotient (36) until you are left with 1.
2
2
2
3
3
72
36
18
9
3
1
All the numbers on the left are the prime factors of 72. The factor tree is helpful, but is
not compulsory.
2. Index Notation
Now we know that 72 can be expressed as follows:
72 = 2 × 2 × 2 × 3 × 3
There a neater way to express the above.
We can express the number of times a number is multiplied by itself with a power, which
is also called an exponent, or index (plural: indices).
3 × 3 = 32 (3 𝑠𝑞𝑢𝑎𝑟𝑒𝑑 )
3 × 3 × 3 = 33 (3 𝑐𝑢𝑏𝑒𝑑 )
3 × 3 × 3 × 3 = 34
So we write our final answer as:
72 = 23 × 32
This is called the index notation of the number 72.
Examples: Complete the following examples.
8 = 2×2×2
= 23
18 = 2 × 3 × 3
= 2 × 32
30 =
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50 =
Page 3 of 11
Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Rules of Indices
A. Multiplying and Dividing
Multiplication
Division
33 × 32 = 3 × 3 × 3 × 3 × 3
= 35
33 ÷ 32 = 31
=3
When a number is multiplied by the same
number with a different power, we add the
powers: 3 + 2 = 5
When a number is divided by the same number
with a different power, we subtract the
powers: 3 – 2 = 5
B. Finding Square Roots and Cube Roots
Square roots (√ ): A square root of a
number Q finds the number when multiplied
by itself twice gives the number Q.
𝑒. 𝑔. 5 × 5 = 25 ; √25 = 5
3
Cube roots (√ ): A cube root of a number
Q finds the number when multiplied by
itself thrice gives the number Q.
3
𝑒. 𝑔. 3 × 3 × 3 = 27 ; √27 = 3
3
√26 = √23 × 23 = 23
To find the square root of a number, we divide
the powers of its prime factors by 2.
3
√26 = √22 × 22 × 22 = 22
To find the cube root of a number, we divide
the powers of its prime factors by 3.
Examples: Complete the examples using the prime factorisation method.
√36 = √22 × 32
= 21 × 31
=6
3
3
√2744 = √73 × 23
= 71 × 21
= 14
√144
=
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3
√216
=
Page 4 of 11
Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Exercise 2: Prime Factorisation
Complete the following questions on a foolscap paper and show your workings.
Basic
1.
Express the following as a product of their prime factors:
a.
b.
2.
153
120
c.
d.
2744
2420
e.
f.
1470
12375
a. Given that 594 = 2𝑥 × 3𝑦 × 11𝑧 . Find the values of 𝑥, 𝑦 and 𝑧.
b. Given that 1568 = 2𝑥+1 × 7𝑦 . Find the values of 𝑥 and 𝑦.
Intermediate
3.
Using the prime factorisation method, find the square roots of the following:
a.
b.
4.
c.
d.
2304
625
e.
f.
7056
48400
Using the prime factorisation method, find the cube roots of the following:
a.
b.
5.
225
784
2744
1728
c.
d.
3375
5832
e.
f.
9261
2195
a. Given that √7.29 = 2.7 and √72.9 = 8.54, find the square root of 72900 without using
the calculator.
3
3
b. Given that √1.05 = 1.02 and √10.5 = 2.19, find the cube root of 10500 without using
the calculator
Mastery
6.
a. Given that 450 = 2 × 32 × 52, state the smallest value of 𝑘 such that 450𝑘 is a perfect
square.
b. Express 252 as a product of its prime factors. Hence, find the smallest value of 𝑘
such that 252𝑘 is a perfect square.
c. Express 726 as a product of its prime factors. Hence, find the smallest value of 𝑘
such that 726𝑘 is a perfect square.
7.
a. Given that 179685 = 33 × 5 × 113, state the smallest value of 𝑚 such that 179685𝑚 is
a perfect cube.
b. Express 280 as a product of its prime factors. Hence, find the smallest value of 𝑚
such that 280𝑚 is a perfect cube
c. Given that 21168 = 24 × 33 × 72, find the smallest value of 𝑚 such that 21168𝑚 is a
perfect cube.
8.
a. Find the index notation of 392. Hence, find the smallest whole number that 392 can
be divided by to obtain a perfect square.
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Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
b. Given that 1296 = 24 × 34, find the smallest whole number that 1296 can be divided
by to obtain a perfect cube.
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Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Section 3: Highest Common Factors
Suppose you have a plain canvas and you wanted to decorate it with some mosaic tiles. The
canvas measures 18 cm by 27 cm, and the shop you go to sells only square tiles but in
varying dimensions. What would be the largest dimension of the tiles you can buy to cover
the whole canvas, and not require you to cut the tiles?
27 cm
To find the dimensions of the tiles, we need to
consider the factors of 18 and 27:
18 cm
Factors of 18 = 2, 3, 6, 9, 18
Factors of 27 = 3, 9, 27
?
Looking at the factors of 18, we can use tiles measuring 2, 3, 6, 9, or 18 cm as there would
be no excess along the 18 cm side. However, amongst these, only a 9 cm tile can fit along
the 27 cm side without excess as 27 is not divisible by the other numbers. So the tile we
should buy is 9 cm, which is the Highest Common Factor of 18 and 27.
The Highest Common Factor (HCF) of two or more numbers is the largest common
factor of these numbers.
Since we can’t always list all the factors of the numbers for which we are finding the HCF,
there are two methods we can use to find HCF for any group of numbers.
Index Notation Method
1. Write the index notation for both numbers.
2. The HCF is the product of the each common prime factor
with the lowest power in both numbers.
18 = 2 × 32
27 = 33
HCF = 32 = 9
Factor Tree Method
1. Divide both numbers repeatedly by only their common
factors. Stop when there are no common factors.
2. The HCF is the product of all common prime factors on the left.
(3 × 3 = 9)
3 18 , 27
3
6, 9
2, 3
Examples: Find the HCF of the following numbers
28 and 42
378 and 504
28 = 22 × 7
42 = 2 × 3 × 7
378 = 2 × 33 × 7
504 = 23 × 32 × 7
HCF = 2 × 7 = 14
HCF = 2 × 32 × 7
= 126
54 and 90
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72 and 108
Page 7 of 11
Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Section 4: Lowest Common Multiples
Now, suppose you have another problem. You need to buy some hot dogs and buns for a
party you are hosting but hot dogs are sold in packs of 10 and buns are sold in packs of 8.
How many packs of each should you buy to so that you don’t have any of either left over?
This time, we need to consider the multiples of 10 and 8, since you will be buying multiple
packs of both.
Multiples of 10 = 10, 20, 30, 40, 50
Multiples of 8 = 8. 16. 24. 32. 40, 48
Looking at the above, we can see that the lowest common multiple of both numbers is 40.
This means that you need to buy 40 hot dogs and 40 buns, which is 10 packs of hot dogs
and 5 packs of buns respectively.
The Lowest Common Multiple (LCM) of two or more numbers is the smallest common
multiple of these numbers.
You can also find LCM of a set of numbers using two methods.
Index Notation Method
1.
2.
Write the index notation for both numbers
The LCM is the product of every prime factor in each
number, with the highest power in both numbers.
10 = 2 × 5
8 = 23
HCF = 23 × 5
= 40
Factor Tree Method
1.
2.
Divide both numbers repeatedly by only their common factors.
You can stop when there are no common factors.
The LCM is the product of the all factors on the left and the
remaining factors below. ( 2 × 5 × 4 = 40)
2 10 ,
5,
8
4
Examples: Find the LCM of the following numbers
36 and 42
324 and 504
36 = 22 × 32
42 = 2 × 3 × 7
324 = 22 × 34
504 = 23 × 32 × 5
LCM = 22 × 32 × 7
= 14
HCF = 22 × 34 × 5
= 3240
54 and 90
72 and 108
HELP! How do you know when the question is asking for LCM or HCF?
Ask yourself if what you are looking for is more or less than the original numbers.
 If it is less than the original numbers, you should find HCF.
 If it is more than the original numbers, you should find LCM.
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Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Exercise 3: HCF and LCM
Complete the following questions on a foolscap paper. Show your workings clearly.
Basic to Intermediate
1.
Find the HCF of the following sets of numbers:
a. 44, 121 and 132
b. 170, 204 and 578
2.
Find the LCM of the following sets of numbers:
a. 20, 70 and 100
b. 132, 156 and 180
3.
c. 23 × 32 × 52 × 7 and 22 × 5 × 7
d. 33 × 5 × 11 and 2 × 112
Find both the LCM and HCF of the following sets of numbers, giving your answers in
index notation:
a. 68, 126 and 154
b. 75, 45 and 125
4.
c. 23 × 52 × 13 and 22 × 53 × 13
d. 22 × 3 × 5 and 24 × 32 × 19
c. 26 x 32 x 5 x 74 and 24 x 3 x 53 x 7
d. 23 × 34 × 52 × 7, 24 × 37 × 5, and 32 × 53 × 112 × 13
a.
A number leaves a remainder of 1 when divided by 3, 6 or 7. What is the least
possible value of the number?
b. A number leaves a remainder of 3 when divided by 8, 10 and 12. What is the least
possible value of this number?
Mastery
5.
Two numbers have a LCM of 120 and a HCF of 2. Find the possible values of the two
numbers.
6.
a. The LCM of 6, 21 and 𝑥 is 126. Find the least possible value of 𝑥.
b. The LCM of 12, 18 and 𝑦 is 504. Find the least possible value of 𝑦.
c. The LCM of 63, 54 and 𝑧 is 3780. Find the least possible value of 𝑧.
7.
a. Find the smallest number which when divided by 7, 8 and 9 leaves a remainder of 4,
5, and 6 respectively.
b. Find the smallest number which when divided by 8, 12 and 15 leaves a remainder
of 6, 10 and 13 respectively.
c. Find the smallest number which when divided by 25, 30 and 40 leaves a remainder
of 20, 25 and 35 respectively.
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Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
Exercise 4: HCF and LCM Word Problems – Basic to Intermediate
Consider carefully whether the following questions require HCF or LCM. Complete them on
a foolscap paper and show your workings clearly.
1.
A floor of a hotel room is to be covered with square tiles of the same size. If the floor
measure 480 cm by 700 cm, find
a. The largest possible length, in cm, of the side of each tile,
b. The number of such tiles requires to cover the whole floor.
2.
Three bells ring at intervals of 8 minutes, 12 minutes and 15 minutes. They first ring at
the same time at 9 am. What time will the three bells next ring together again?
3.
Alicia, Jane and David visit the school library every 6 days, 2 weeks and 15 days
respectively. If all three of them meet one another on a particular day, how many weeks
later will all three of them next meet each other again?
4.
A shopkeeper has 48 apples, 72 oranges and 96 pears in his storeroom. If each type of
fruit is distributed equally among a certain number of fruit baskets, what is the greatest
number of fruit baskets that can be prepared?
5.
Jill wants to make a square using rectangular cards measuring 14 cm by 25 cm. What is
the least number of cards she needs?
6.
The Earth, Jupiter, Saturn and Uranus take about 1 year, 12 years, 30 years and 84
years respectively to revolve around the Sun.
a. How often will you get a planetary alignment of the Earth, Jupiter and Saturn?
b. Jupiter, Saturn and Uranus first aligned in 1978. Which year will they next align
again?
7.
A school needs to pack 378 chocolates and 441 sweets into identical packets to be
distributed to as many children as possible.
a. How many packets can there be?
b. How many chocolates and sweets are there in each packet?
8.
Clara bought three rolls of ribbons of different lengths and colours. The blue ribbon is 60
cm long, the red ribbon is 84 cm long and the green ribbon is 108 cm long. She intends
to cut them all into equal length without wastage.
a. What is the longest possible length of each piece?
b. After cutting, how many pieces of ribbons will she have?
9.
Trees, benches and lamp posts are found at 15 m, 20 m and 25 m intervals along a
road respectively. At the beginning of the road, all three objects are in line.
a. At what distance from the beginning of the straight road will the three objects next
appear in line?
b. If the straight road is 2 km long, how many times, including the first time, do the
objects appear in line?
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Secondary 1 Mathematics
Topic 1 – Prime Numbers and Prime Factorisation
10.
Three long-haul trains leave a depot at intervals of 14 mins, 42 mins and 𝑥 minutes
respectively. If they first left together at 7 am, and next left together again at 5.30 pm,
what is the smallest value of 𝑥?
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