DMS1A_Exp_Chp1 (2)

1A

Chow Wai Keung
General Editor: Ann Lui Yin Leng
Consultant: Prof Ling San

1 MFauclttoiprlseAsnd

Let’s Learn to

recognise prime numbers

Most of our vehicle number express a composite number as a product of prime factors

plates consist of a 3-letter represent the prime factorisation of a number in index notation
prefix, followed by 3 or 4 digits
and finally a checksum letter. f ind the highest common factor (HCF) and lowest common
Do you know which letters multiple (LCM) of a group of numbers by using prime
never appear at the end? This factorisation
checksum letter is generated
by the second and third letters u nderstand the meaning of the square root and cube root
of a number

of the prefix, and the digits on
the number plate. The letter
generation involves the use of
a prime number.

1.1 Primes, Prime Factorisation And Index MATHS WEB
Notation
You may visit the website
Let us revise what we have learnt about factors and multiples at the primary level. http://www.starpub.com.
sg/dm/s1e/weblinks_1a.
A Factors html to have more practice
on factors.
0, 1, 2, 3, 4, … are A whole number
whole numbers. They greater than 1 can be
can be used to count written as a product
things. of two whole numbers.

The number 12 can be expressed as products of two whole numbers as follows:
12 = 1 × 12,
12 = 2 × 6, 
12 = 3 × 4. 
1, 2, 3, 4, 6 and 12 are called factors of 12. Notice that 12 is divisible by each of its factors.

E xample 1 Find the factors of 70. REMARK

Students should check and
  Solution 70 = 1 3 70 verify that other numbers
like 3, 4, 6, 8, 9, ... between
= 2 3 35 1 and 70 are not factors
= 5 3 14 of 70.
= 7 3 10

The factors of 70 are 1, 2, 5, 7, 10, 14, 35 and 70.

Try It! 1 Find the factors of 105.

B Multiples

When a number is multiplied by a non-zero whole number, we get a multiple of the
number.
The multiples of 3 are  3 3 1, 3 3 2, 3 3 3, 3 3 4, 3 3 5, …
i.e. 3, 6, 9, 12, 15, …

Do you see the link between factors and multiples?
Recall that 3 and 4 are factors of 12. On the other hand, 12 is a multiple of both 3 and 4.

2 Chapter 1  Factors And Multiples

C Primes

1

Objective: To classify whole numbers based on their number of factors.

Tasks

1. Do the following task with a partner. Consider the following whole numbers, list their factors in the table
below.

Number Factors Number Factors

2 12
3 13
4 14
5 15
6 16
7 17
8 18
9 19
10 20
11 21


2. Classify each of the above numbers according to the number of factors they have. You may want to use the
table below to help you organise.

Number of factors Numbers RECALL

1 • Multiples of 2 are
2 2, 4, 6, 8, ...
3 • Multiples of 3 are
4 3, 6, 9, 12, ...
5 • Multiples of 5 are
More than or equal to 6 5, 10, 15, 20, ...



3. Observe those numbers that have only two factors. List them out. What patterns do you observe about the
factors of these numbers? The numbers that you have listed out have a special name. Do you know what
they are called?

4. Let us examine the numbers 0 and 1. How many factors do they have?

3

The numbers you have just listed out in Class Activity 1 are called prime numbers. REMARK
A prime number is a whole number greater than 1 that has only
two factors, 1 and itself. A prime number is also
called a prime.
Notice that the first 10 prime numbers are 2, 3, 5, 7, 11, 13, 17, 19, 23 and 29.
The number 6 which has four factors 1, 2, 3 and 6 is called a composite number. DISCUSS

A composite number is a whole number greater than 1 that has 1. Explain why 0 and 1 are
more than two factors. not prime numbers.

0 and 1 are neither prime numbers nor composite numbers. 2. Can a whole number
greater than 1 be both
a prime number and a
composite number?

E xample 2 Show and explain whether the following are prime or composite DISCUSS
numbers.
In practice, it is not
(a) 851 (b) 113 necessary to test the
divisibility of all the primes
  Analysis If a number is divisible by a composite number (e.g. 18), then it is smaller than the given
divisible by the prime numbers which are the factors of the composite number. Do you know why?
number (e.g. 2 and 3). Therefore, to determine whether a number is a
prime, we can check if it is divisible by the prime numbers smaller than MATHSmatter
the given number.
Sir Henry Billingsley first
  Solution (a) By going through the division of 851 by the prime numbers 2, 3, translated Euclid’s Elements
in Greek into English in
5, ... one by one, we find that 1570. The word ‘protos’ in
the Elements was translated
851 ÷ 23 = 37 851 is divisible by 23. as ‘prime’. The word ‘prime’
is from the Latin word
i.e. 851 = 23 3 37 You can use your calculator to verify the ‘primus’ which means first.

answer. MATHSmatter

∴  851 is a composite number. It can be proved that
there are infinite numbers
of primes. Large prime
(b) 113 is not divisible by all the primes 2, 3, 5, 7, ... less than 113. It numbers are used for the
encryption of messages
is only divisible by 1 and itself. and data on the Internet.

∴  113 is a prime number.

Try It! 2 Show  and explain whether the following are prime or composite numbers.
(a) 127 (b) 473

4 Chapter 1  Factors And Multiples

D Prime Factorisation

A prime number which is a factor of a composite number is called a prime factor of the
composite number. For example,

30 = 2 × 3 × 5
{
{
{
{
composite

number prime factors of 30

The way to express a composite number as a product of prime
factors only is called prime factorisation.

We can use a factor tree to find the prime factorisation of a composite number as illustrated
in the following example.

E xample 3 Find the prime factorisation of 1020.


  Solution We can build a factor tree as follows:

1 Write down two numbers 2 Continue to factorise, if

whose product is 1020, possible, 2 and 510.

e.g. 1020 = 2 3 510, as 1020
follows:

   1020 2 510
10 51
REMARK
2 510
We can also find the prime
factorisation of a number by
3 We stop when the last row of the tree shows the prime factors. The successive short division of
product of all the prime factors in the tree is the prime factorisation the number by its prime
of the given number. factors until the quotient is
reduced to 1.
1020
e.g.  2 1020
2 510 2 510
3 255
10 51 5 85
17 17
2 5 3 17 1

∴  1020 = 2 3 2 3 3 3 5 3 17 ∴ 1020 = 2 3 2 3 3 3 5 3 17

Try It! 3 Find the prime factorisation of 585 using a factor tree and compare it i.e.  t he prime factorisation
with those of your classmates. is the product of all the
divisors.

5

Example 4 Find the prime factorisation of 3135.

  Solution We can also divide the number 3135, first by its smallest prime factor 3. REMARK

Then, by its successive prime factors until we find all its prime factors A number is divisible by
Try It! 4 3 if the sum of its digits
as shown below. 3135 is divisible by 3.

3 1045 A number is divisible by
5 if its last digit is 0 or 5.
5 209
11 19

∴  3135 = 3 3 5 3 11 3 19

Find the prime factorisation of 2730.

E Index Notation

When a number is multiplied by itself more than once, we can use a notation to represent
the product as shown below.

6 3 6 = 62, read as 6 squared or 6 to the power of 2;
6 3 6 3 6 = 63, read as 6 cubed or 6 to the power of 3;
6 3 6 3 6 3 6 = 64, read as 6 to the power of 4, and so on.

This notation is called the index notation. In this notation, the number 6 to the power of 4 index
at the level is called the base and the number at the top right-hand
corner is called the index. The index shows the number of times the 64
base is multiplied by itself.
base
We can use an index notation to represent the prime factorisation of a number.
REMARK
Example 5 Find the prime factorisation of 792, expressing the answer in index
notation. Alternative method:
  Solution   2 792
We can use the factor tree method to find the factors of 792. 2 396
792 2 198
3 99
8 99 3 33
11 11
2 4 9 11 1

2 23 3 ∴  792 = 232323333311
∴  792 = 2 3 2 3 2 3 3 3 3 3 11 = 23 3 32 3 11

= 23 3 32 3 11

Try It! 5 Find the prime factorisation of 702, expressing the answer in index
notation.

6 Chapter 1  Factors And Multiples

EXERCISE 1.1

LEVEL 1 12. Find the prime factorisation of the following

numbers, expressing your answers in index

1. Write down all the factors of each of the following notation.
numbers.
(a) 180 (b) 616

(a) 15 (b) 28 (c) 735 (d) 1350

(c) 32 (d) 43 LEVEL 2

13. (a) List all the factors of 56.
2. Write down the first four multiples of each of the (b) List all the factors of 84.
(c) Hence, find all the factors common to 56
following numbers.
and 84.
(a) 2 (b) 5

(c) 11 (d) 23 14. (a) Write down the first ten multiples of 2.
(b) Write down the first ten multiples of 3.
(c) Hence, write down the first three multiples
3. Determine whether 7 is a factor of 2395.
common to 2 and 3.
4. Determine whether 2816 is a multiple of 11. (d) What can you say about the numbers in (c)?

5. Find the smallest prime factor of 377.
15. Determine whether each statement below is
6. Find the largest multiple of 17 which is less than
1000. true or false. Give a specific counterexample for
those that are false.
7. Find the smallest multiple of 19 which is greater (a) If 6 is a factor of a number, then 3 is a factor
than 500.
of the number.
8. Determine whether the following numbers are (b) If 2 and 7 are factors of a number, then 14 is

prime numbers. a factor of the number.
(c) If 2 and 8 are factors of a number, then 16 is
(a) 103 (b) 229 (c) 817
a factor of the number.
9. Express the following in index notation.
(a) 8 3 8 3 8 16. Determine whether each statement below is true
or false. Give a specific counterexample for those
(b) 3 3 3 3 3 3 3 3 3
(c) 7 3 7 3 9 that are false.

(d) 4 3 4 3 6 3 6 (a) If two numbers are multiples of 11, then
(e) 2 3 3 3 11 3 11 3 11 their sum is a multiple of 11.

(f) 5 3 5 3 13 3 5 3 13 3 37 (b) If a number is a multiple of 2 and another
number is a multiple of 3, then their sum is a
10. Find the values of the following.
multiple of 5.
(a) 172 (b) 53

(c) 22 3 112 (d) 34 3 25 17. (a) Find the missing

11. Find the prime factorisation of the following numbers in the factor

numbers using the factor tree method. Express tree. 3
(b) Can you find the number
your answers in index notation.
at the top of the tree 11
(a)
54 (b) 72 without finding the other

two numbers? Explain 2 7

64 briefly.



7

18. Express each of the following as a single number 23. Is it true that if a number is a multiple of 2 and
another number is a multiple of 3, then their
in index notation. product is a multiple of 6? Explain briefly using
an example.
(a) 23 3 22 (b) 34 3 35

(c) 115 ÷ 112 (d) (53)2

19. Express each of the following in index notation, 24. The prime numbers 17 and 19 are called twin
primes because they differ by 2. List three other
where all the bases are prime numbers. pairs of twin primes.

( a) 12 3 15 (b) 54 3 98

(c) 18 3 10 3 75 (d) 33 3 60 3 125

25. Express each of the following even numbers as a

LEVEL 3 sum of two prime numbers.

(a) 32 (b) 78 (c) 116

20. The Olympic Games are Note: A mathematician proposed that “Every
held in years that are
multiples of 4. In which even number greater than 2 can be
years will the next three
expressed as a sum of two prime numbers.”
Olympic Games be held after the London Olympic
Games in 2012? Do you agree? Why? This proposal is

called Goldbach Conjecture. You may

surf the Internet to find more information

about it.

21. A design is formed by arranging a certain number 26. The prime factorisation of a number is

of square tiles to form a rectangular array. Find 24 3 35 3 72 3 11.

the number of possible distinct designs if there Write down 3 factors of the number that are
greater than 100.
are
Hint: 16 × 11 = 176 which is a factor greater than
(a) 18 tiles, (b) 41 tiles. 100.

22. A lock can only be opened by using a 3-digit 27. The prime factorisation of two numbers are
number. Jinlan sets this 3-digit number to be the
largest prime number less than 1000. What is this 2 3 32 3 73 3 13 and 3 3 72 3 133 3 17.
number?
A common factor of these two numbers is
3 3 7 = 21. Write down 3 other common factors of
these two numbers

1.2 Highest Common Factor (HCF) REMARK

Let us consider the factors of 18 and 24. HCF is also known as
Factors of 18: 1, 2, 3, 6, 9, 18 greatest common divisor
Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 (GCD).
1, 2, 3 and 6 are the common factors of 18 and 24. Among them, the largest common factor
is 6. We say that 6 is the highest common factor (HCF) of 18 and 24. MATHSmatter

The highest common factor (HCF) of two or more positive whole numbers is When two numbers, such
the largest positive integer that divides the numbers without a remainder. as 15 and 16, have no
common factors greater
There are different methods of finding the HCF of two or more numbers. We can use prime than 1, their HCF = 1 and
factorisation to find their HCF in a more efficient way. the numbers are said to be
relatively prime or coprime.
8 Chapter 1  Factors And Multiples

E xample 6 Find the HCF of 225 and 750 using prime factorisation. REMARK

  Solution We find the prime factorisation of each number. Alternative method:
We use short division to
225 = 32 3 52 divide both numbers by
750 = 2 3 3 3 53 their common factors
successively until there are
We can visualise the process of taking common factors as shown. no common prime factors.

225 = 33 3 3 5 3 5 For example,
750 = 2 3 3 35 35 35
3 225 750
HCF = 3 35 35 5 75 250
5 15 50
∴  HCF = 3 × 52 3 is of a lower power than 32 and
= 75 52 is of a lower power than 53. 3 10

Note: We see that the HCF is obtained by multiplying the lowest power HCF = 3 3 5 3 5
of each common prime factor (3 and 52) of the given numbers. = 75
i.e.  the HCF is the product
Try It! 6 Find the HCF of 252 and 360.
of all the divisors.
Example 7 Find the HCF of 84, 126 and 245.
MATHS WEB
  Solution 84 = 22 3 3 3 7
Another efficient method
126 = 2 3 32 3 7 of finding the HCF of two
245 = 5 3 72 numbers is called Euclidean
Algorithm. Visit the website
7 is the only prime factor common to all the three numbers. at http://www.starpub.com.
∴  HCF = 7 sg/dm/s1e/weblinks_1a.
html for more information
on Euclidean Algorithm.

REMARK

Try It! 7 Find the HCF of 154, 330 and 396. Alternative method:

7 84 126 245
12 18 35

Example 8 A rectangular field of 20 m by 15 m is divided into identical square plots ∴  HCF = 7
of land. Find the largest possible length of a side of each square plot of land.

  Solution The length (in metres) of a side of each square 5
plot of land should divide 20 and 15 exactly. 5

Hence, it is a common factor of 20 and 15. 15
Therefore, the largest possible length is the HCF
of 20 and 15.

20 = 22 3 5 20
15 = 3 3 5 The figure shows the division of the land.
∴  HCF of 20 and 15 is 5.

The largest possible length of a side of each square plot of land is 5 m.

Try It! 8 A rectangular piece of paper of 35 cm by 28 cm is cut to obtain identical
squares. Find the largest possible length of a side of each figure.

9

EXERCISE 1.2 6. The total sales of a particular model of calculator
on three days are $180, $414 and $306 respectively.
LEVEL 1 Find the greatest possible price of the calculator.

1. Find the HCF of each pair of numbers. 7. A charitable organisation distributes 420 packets
(a) 8 and 12 of rice, 168 bottles of oil and 504 oranges equally
among some senior citizens. Find the greatest
(b) 18 and 27 possible number of the senior citizens.
(c) 45 and 72
8. A botanist conducts a controlled experiment
(d) 21 and 84 using 48 seeds A and 78 seeds B. Each type of
(e) 74 and 99 seeds is divided equally into as many groups as
possible such that the number of groups of seeds
(f) 120 and 225 A and the number of groups of seeds B are the
(g) 108 and 240 same. Find

(h) 385 and 396 (a) the number of groups of seeds A,
(b) the number of seeds A in each group,
LEVEL 2 (c) the number of seeds B in each group.

2. Find the HCF of each group of numbers. 9. The product of two numbers is 8820. The HCF of
(a) 28, 63 and 91 these numbers is 42. Find the greater number.

(b) 60, 75 and 100 10. Find two different numbers such that their HCF is
(c) 48, 84 and 144 18.

(d) 66, 110 and 847 11. Find three numbers such that the HCF of each
(e) 14, 36 and 175 pair of these numbers is greater than 1 but the
HCF of all three numbers is 1.
(f) 70, 210 and 350
Hint: F or instance, the numbers 6, 10 and 15
LEVEL 3 satisfy the conditions.

3. There are two metal bars of lengths 72 cm and 12. Find the greatest number that will divide 171, 255
96 cm. Both bars are exactly cut into small pieces and 304 so as to leave the same remainder in each
of equal lengths. Find the largest possible length case.
of each small piece.
13. The price (in dollars) of a model car in a shop is
4. A rectangular piece of tin plate measuring a whole number greater than 1. The sales of the
360 cm by 280 cm is cut into small identical model cars on two days are $1518 and $2346.
squares of the largest possible length. Find How many model cars are sold on each day?

(a) the length of a side of each square,
(b) the number of square tin plates formed.

5. 126 pieces of white chocolate and 84 pieces of
dark chocolate are divided into packs with equal
number of pieces of each kind of chocolate. What
is the greatest possible number of packs needed?

10 Chapter 1  Factors And Multiples

1.3 Lowest Common Multiple (LCM)

Let us consider the first 10 multiples of 6 and 8.
Multiples of 6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, …
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, …

What do you notice Some of the multiples
about the multiples are common to both
of both numbers? numbers.

24 and 48 are the common multiples of 6 and 8. The smallest common multiple of 6 and 8
is 24. We say that 24 is the lowest common multiple (LCM) of 6 and 8.

The lowest common multiple (LCM) of two or more whole numbers is the
smallest common multiple of the numbers.

We can use prime factorisation to find the LCM of two or more numbers as shown by the RECALL
following examples.
We can use prime
\E xample 9 Find the LCM of 24 and 90. factorisation to find the HCF
of two or more numbers.
  Solution We find the prime factorisation of each number.
REMARK
24 = 23 3 3
90 = 2 3 32 3 5 Alternative method:
We can visualise the process of identifying all the different factors as Carry out successive short
divisions by common
shown below. factors.

33 2 24 90
24 = 2 3 2 3 2 33 3 3 3 5 3 12 45

90 = 2 4 15
LCM
LCM = 2 3 2 3 2 3 3 3 3 3 5 = p roduct of the common

∴  LCM = 23 3 32 3 5 23 is of a higher power than 2 and factors and the quotients
= 360 32 is of a higher power than 3. = 2 3 3 3 4 3 15
=23332323335
Note: We see that the LCM is obtained by multiplying the highest power = 23 3 32 3 5
of each prime factor (23, 32 and 5) of the given numbers. = 360

Try It! 9 Find the LCM of of 40 and 150.

11

Example 10 Find the LCM of 26 and 99.



  Solution 26 = 2 3 13

99 = 32 3 11

∴ LCM = 2 3 32 3 11 3 13
= 2574

Note: When two numbers are relatively prime, they have no common REMARK
factors greater than 1. Their LCM is the product of the two
numbers as in the above example (26 3 99 = 2574). Alternative method:
Carry out successive short
Find the LCM of 34 and 57. divisions by common
Try It! 10 factors for at least two
numbers. If a number is not
Example 11 Find the LCM of 20, 24 and 70. divisible by the factor, copy
the number to the next row.
  Solution 20 = 22 3 5
2 20 24 70
24 = 23 3 3
70 = 2 3 5 3 7 2 10 12 35
∴  LCM = 23 3 3 3 5 3 7
= 840 5 5 6 35

Try It! 11 Find the LCM of 54, 84 and 110. 167

LCM
=232353637
=23235323337
= 23 3 3 3 5 3 7
= 840

Example 12 aTrhee6c0irccmumanfedr4en5ccems of the front wheel and the rear wheel of a tricycle
respectively. When Bob begins to ride the tricycle,

point P on the front wheel and point Q on the rear wheel touch the

ground.

(a) What is the distance travelled before P and Q next touch the

ground at the same time? P

(b) Find the numbers of revolutions that the front and rear wheels

each will have made by then. Q

  Analysis Point P touches the ground again when the front wheel makes one

revolution, that is when the tricycle moves by 60 cm. Point Q touches
the ground again when the rear wheel makes one revolution, that is when
the tricycle moves by 45 cm. Therefore, the distance, in centimetres,
required is the LCM of 60 and 45.

  Solution (a) 60 = 22 3 3 3 5

45 = 32 3 5

∴ LCM = 22 3 32 3 5
= 180

The distance travelled is 180 cm.

12 Chapter 1  Factors And Multiples

(b) 180 ÷ 60 = 3
The front wheel has made 3 revolutions.
180 ÷ 45 = 4
The rear wheel has made 4 revolutions.

Note: We can illustrate the horizontal distances covered by the two
wheels as shown below.

Front Wheel: 60 cm P1 60 cm P2 60 cm P3
P Q4
45 cm Q1 45 cm Q2 45 cm Q3 45 cm
Rear Wheel:
Q

Try It! 12 The figure shows a gear system in
which the numbers of teeth on the big and

small wheels are 20 and 16 respectively. X
The tooth X on the big wheel and the Y

tooth Y on the small wheel are engaged

at the start.

(a) Find the number of tooth contacts that the two wheels will make

before X and Y are engaged again.

(b) Find the number of revolutions that each wheel will have made by

then.

Example 13 For the celebration of the year of rabbit in the Chinese calendar in
2011, Singapore Post issued a series of Rabbit Zodiac stamps. Their
denominations and sizes are as follows:

Denominations: 1st Local, 65 cents and $1.10
Stamp Size: 40 mm 3 30 mm (1st Local and 65 cents)
35 mm 3 45.65 mm ($1.10)

Siti placed one row of 65-cent stamps and one row of $1.10 stamps in
her album as shown below. Find the minimum number of stamps in
each row such that the two rows are of the same length.



13

  Solution 40 = 23 3 5

35 = 5 3 7

The minimum length, in mm, of each row = LCM of 40 and 35

= 23 3 5 3 7

= 280

Hence, number of 65-cent stamps = 280 ÷ 40

= 7

and number of $1.10 stamps = 280 ÷ 35

= 8

Try It! 13 The dimensions of a 1st Local stamp are 40 mm 3 30 mm. Wei Ming
placed some 1st Local stamps in two rows in different orientations as
shown below. Find the minimum number of stamps in the first row
such that the two rows are of the same length.



EXERCISE 1.3

LEVEL 1 4. (a) Find the HCF and LCM of 18 and 30.
(b) Find the HCF and LCM of 14 and 35.
1. Find the LCM of each pair of numbers. (c) What is the relationship between the

( a) 12 and 15 (b) 6 and 28 product of the HCF and LCM of two
numbers and the numbers themselves?
( c) 25 and 40 (d) 23 and 32 (d) The HCF of two numbers is 13 and their
LCM is 3640. If one number is 520, find the
(e) 24 and 54 (f) 60 and 75 other number.

(g) 59 and 118 (h) 65 and 91 LEVEL 3

LEVEL 2 5. A flash bulb on a Christmas tree flashes once
every 10 seconds. Another bulb flashes once every
2. Find the LCM of each group of numbers. 15 seconds. If they are flashing together now, how
long will it take for the two bulbs to flash together
(a) 9, 12 and 30 (b) 13, 14 and 15 next?

(c) 6, 8 and 20 (d) 28, 42 and 105 6. John and Arun are running along a circular track.
They take 48 seconds and 56 seconds respectively
(e) 22, 132 and 253 (f ) 4, 9 and 31 to complete a lap. They begin from the same
point at the same time in the same direction.
3. The prime factorisation forms of two numbers are
24 3 35 3 53 3 72 and 23 3 36 3 5 3 78. Find (a) How long does it take for them to meet at
the starting point again?
(a) the HCF of these two numbers in prime
factorisation form, (b) How many laps will each boy have run by
then?
(b) the LCM of these two numbers in prime
factorisation form.



14 Chapter 1  Factors And Multiples

7. The thickness of a Science book is 20 mm and 10. Doris has piano tutoring once every 6 days,
that of a Mathematics book is 28 mm. Books of swimming lessons once every 4 days and ballet
each type are stacked up in a separate pile. lessons once every 8 days. If she has all these three
activities on 1 April, on which date will she have
(a) What should be the minimum height of all of them together next?
each pile such that both piles are of the same
height? 11. Buses A, B and C are at a bus interchange at
intervals of 20 minutes, 30 minutes and 45 minutes
(b) Find the number of books in each pile. respectively. If all of them are at the interchange
at 08 30, find the time when they will meet at the
8. Each student desk in a classroom has a rectangular interchange next.

desktop that measures 60 cm by 45 cm. Some of 12. Find three pairs of numbers such that the LCM of
each pair of numbers is 24.
these desks are arranged, side by side, as shown in
13. Find two possible pairs of numbers such that the
the diagram to form a large square table for a class HCF and LCM of each pair of numbers are 21 and
630 respectively.
activity. Find 60
(a) the least length of a 14. Find the greatest 3-digit number which is divisible
side of the square, 45 by 15, 20 and 24.

(b) the number of rows 15. Find the smallest number that should be added
to 1628 so that the sum is exactly divisible by 4,
and columns of desks 5 and 6.

used to form the

large square table.

9. A jewellery cabinet is monitored by three
security cameras. The first camera scans it every
30 seconds, the second scans every 24 seconds
and the third scans every 50 seconds. The three
cameras scan the cabinet together at 06 00. At
what time will they scan the cabinet together
next?

1.4 Square Roots And Cube Roots

A Square Roots 32 = 9 REMARK
32 is read as
Recall that 3 × 3 = 32 3 squared. Students will learn more
when expressed in about positive and negative
index notation. numbers in Chapter 2.

Since 32 = 9, 9 is called the square of 3. We also say that 3 is the positive square root of 9 RECALL
and it is denoted by 9 = 3.
If n is a whole number,
then n2 = n.

Similarly, we write and  4 = 22 = 2, REMARKS
22 = 2 3 2 = 4 and 16 =
and 25 = 42 = 4, CWheeccakn: use the key
42 = 4 3 4 = 16 52 = 5. Wonetchaencuaslceutlahteor to finkdeythe
osnqutahreecraoloctuolaftoarntuomfibnedr.the
52 = 5 3 5 = 25

The numbers 1, 4, 9, 16, 25, … whose square roots are whole numbers are called perfect square root of a number.
squares. We can find the square root of a perfect square by using prime factorisation.

15

E xample 14 Find the value of 144 . REMARK

  Solution Express 144 in prime factors. Observe that the index in each
of the following values, 24 and
144 = 24 3 32 32, is a multiple of 2.
Therefore, we can split
Split the prime factors into two equal groups. 24 3 32 into two equal groups,
i.e. (22 3 3) 3 (22 3 3).
144 = (22 3 3) 3 (22 3 3)  Write the RHS as the square of
= (22 3 3)2 a n umber. REMARK
2
∴ 144 == 1222 3 3  3= Check:
= 22 3 3.   Press 144 =  .
Result display: 12
Here,

Take the square root.

Try It! 14 Find the value of 484 .

E xample 15 The area of a square is 1521 cm2. Find the length of a side of the square. REMARK

  Solution Area of a square = Length 3 Length Observe that the index in
each of the following values,
∴  length of the square = area 1521 cm2 ? 32 and 132, is a multiple of 2.
1521 = 32 3 132 ? Therefore, we can split
32 3 132 into two equal groups,
= (3 3 13) 3 (3 3 13) i.e. (3 3 13) 3 (3 3 13).

= (3 3 13)2

∴  1521 = 3 3 13

= 39

The length of a side of the square is 39 cm.

Try It! 15 The area of a square is 7225 cm2. Find the length of a side of the square.

B Cube Roots RECALL

We can express 8 as a product of three identical numbers as follows: If n is a whole number,
2 3 2 3 2 = 8. then 3 n3 = n.

We say that 8 is the cube of 2 and 2 is the cube root of 8 which is denoted by REMARK
3 8 = 2.
Similarly, we write We can use the key 3 on
13 = 1 3 1 3 1 = 1 and 3 1 = 3 13 = 1, a calculator to find the cube
33 = 3 3 3 3 3 = 27 and 3 27 = 3 33 = 3, root of a number. You may
43 = 4 3 4 3 4 = 64 and 3 64 = 3 43 = 4. need to press the key SHIFT
first to operate the function.
The numbers 1, 8, 27, 64, … whose cube roots are whole numbers are called perfect cubes.
We can find the cube root of a perfect cube by using prime factorisation.

16 Chapter 1  Factors And Multiples

Example 16 Find the cube root of 216. REMARK

  Solution Express 216 in prime factors. Observe that the index in
each of the following values,
216 = 23 3 33 23 and 33, is a multiple of 3.
Therefore, we can split
Split the prime factors into three equal groups. 23 3 33 into three equal groups,
i.e. (2 3 3) 3 (2 3 3) 3 (2 3 3).
216 = (2 3 3) 3 (2 3 3) × (2 3 3) Write the RHS as the cube of a
number. REMARK
= (2 3 3)3
3 Check:
∴  3 216 == 2 3 3  3 3 =   Press 3 216 =  .
6 Result display: 6
Here, = 2 3 3.

Try It! 16 Find the cube root of 1000. Take the cube root.

E xample 17 The volume of a cube is 3375 cm3. Find the length
of a side of the cube.

  Solution Volume of a cube = Length 3 Length 3 Length

∴ length of the cube = 3 volume ?
3375 = 33 3 53 ?

= (3 3 5) × (3 3 5) 3 (3 3 5) 3375 cm3
= (3 3 5)3 ?
∴  3 3375 = 3 3 5
= 15

The length of a side of the cube is 15 cm.

Try It! 17 The volume of a cube is 2744 cm3. Find the length of a side
of the cube.


EXERCISE 1.4

LEVEL 1

1. Find the value of each of the following numbers. 2. Find the cube root of each of the following

(a) 36 (b) 121 numbers.

(c) 196 (d) 256 (a) 343 (b) 512

(e) 441 (f) 676 (c) 729 (d) 1331

(e) 4096 (f ) 8000

17

LEVEL 2 11. (a) Find the prime factorisation of 129 600.
(b) In an experiment, the speed of a bullet is
3. Find the positive square roots of the following
numbers given in prime factorisation form. found to be 129 600 m/s. Find the value
Express your answers in index notation. of the speed of the bullet in index notation.

(a) 54 3 72 12. The volume of a piece of glass cube is 1728 cm3.
(b) 26 3 1110 Find the length of a side of the cube.

4. Find the cube roots of the following numbers 13. A piece of wire is cut and soldered to form the
given in prime factorisation form. Express your framework of a cube. If the volume of the cube is
answers in index notation. 10 648 cm3, find

(a) 23 3 196 (a) the length of a side of the cube,
(b) 312 3 59 (b) the length of the wire used.

5. (a) Find the HCF of 63 and 117. 14. (a) Find the prime factorisation of 21 952.
(b) Find the positive square root of the HCF (b) The radius of a snowball is found to be
3 21 952 mm. Find the value of the radius in
in (a).
index notation.
6. (a) Find the LCM of 24 and 108.
(b) Find the cube root of the LCM in (a). 15. It is given that 6 is a factor of 5 ◆ 32, where ◆
represents a missing digit.
7. (a) Find the cube of 24 × 52.
(b) Find the positive square root of the result (a) Find all the possible values of ◆.
(b) If 5 ◆ 32 is a perfect cube, find
in (a). (i) the value represented by ◆,
Express your answers in index notation. (ii) the cube root of the number.
(c) Study all the possible numbers 5 ◆ 32
8. (a) Find the square of 76 × 193.
(b) Find the cube root of the result in (a). formed in (a). State some of their common
Express your answers in index notation. properties.

LEVEL 3

9. The area of a square tin plate is 7056 cm2. Find the
length of a side of the plate.

10. The area of a square frame is 2601 cm2. Find the
perimeter of the frame.

18 Chapter 1  Factors And Multiples

utshell Whole Numbers
N 0, 1, 2, 3, …
0, 1
In a Prime Numbers Composite Numbers
2, 3, 5, 7, … 4, 6, 8, 9, …
(neither prime
(has only two factors, (has more than
nor composite) 1 and itself ) two factors)


Factors and Multiples Index Notation
As 24 = 3 × 8, 3 and 8 are factors of 24,
and 24 is a multiple of 3 as well as a multiple of 8. index

8 to the power of 3 83 = 8 × 8 × 8

Prime Factorisation base
The way of expressing a composite number as The index shows the number of times a base is
a product of prime factors only is called prime multiplied by itself.
factorisation.
HCF and LCM
For example, in expressing 340 in prime factors, HCF: Highest Common Factor
we can use a factor tree as follows: LCM: Lowest Common Multiple
We can find the HCF and LCM by prime
340 factorisation.
For example, 700 = 22 3 52 3 7
prime factor 2 170 840 = 23 3 3 3 5 3 7

10 17 prime factor
2 5 prime factors All bases are prime.
HCF = 22 3 5 3 7 = 140
in last row LCM = 23 3 3 3 52 3 7 = 4200

∴  340 = 2 3 2 3 5 3 17

Square and Square Root Cube and Cube Root
Square of 3 = 32 Cube of 5 = 53 = 125 
= 9 Cube root of 125 = 3 125 = 3 53 = 5
32 = 3 A number whose cube root is a whole number is
Positive square root of 9 = 9 called a perfect cube.
= 32
=3 For example, 1, 8, 27, 64, 125, ... are perfect cubes.

A number whose square root is a whole number is
called a perfect square.
For example, 1, 4, 9, 16, 25, ... are perfect squares.

19

REVISION EXERCISE 1

1. Find the smallest number that has 2, 5 and 7 as 7. (a) Find the prime factorisation of the following
its prime factors. numbers in index notation.

2. Find the prime factorisation of the greatest 3-digit (i) 12
number. (ii) 144
(iii) 5040
3. Determine whether each number is prime or (b) The HCF and LCM of two numbers are 12

composite. and 5040 respectively. If one of the numbers
is 144, find the other number.
(a) 649 (b) 721
8. (a) Express 240 as a product of its prime
4. Determine whether each statement below is true factors.
or false.
(b) Find the HCF of 75 and 240.
(a) If 3 and 5 are factors of a number, then 15 is (c) Each box of lollies has 240 pieces. Each pack
a factor of the number.
of cookies has 75 pieces. Mrs Tan buys the
(b) If 246 is a multiple of a number, then 123 is same number of lollies and cookies. Find
a multiple of the number. the least number of boxes of lollies that she
would have bought.
5. (a) Complete the following factor trees.
9. The product of two numbers is 3388 and their HCF
(i) 150 is 11. Find all the possible pairs of the numbers.

25 10. The dimensions of a rectangle are (25 3 7) cm by
(2 3 52 3 73) cm.
(ii)
(a) Find the area of the rectangle. Express your
3 answer in prime factorisation form.

54 (b) A square has the same area as the rectangle.
Find the length of a side of the square.
2
11. (a) Find the value of 2601 .
(b) Write down the prime factorisation in index (b) Find the value of 3 3753 243 .
notation of the number at the top in each tree. (c) Find the HCF of the numbers in (a) and

(c) Find the HCF of the numbers in (b). (b).
(d) Find the LCM of the numbers in (b).
12. A bell rings every 25 minutes while another bell
6. Given three numbers 12, 40 and 45, find rings every 40 minutes. If the two bells rang
(a) their HCF, together at 6 a.m., when will they ring together
(b) their LCM, next?
(c) the greatest 4-digit number which is a
13. Ahmad, Brian and Clara go to ABC Restaurant for
common multiple of these numbers. dinner on regular intervals of 6 days, 8 days and
15 days respectively. If they go to the restaurant
on 1 January, what is the next date that all of
them will go there?

20 Chapter 1  Factors And Multiples

14. A box contains an assortment of 3 types of 16. The Singapore Flyer is a giant observation wheel
chocolate bars. It has 18 bars with almonds, with seating capacity of 784 passengers. The
24 bars with hazelnuts and 30 bars with peanuts. number of passengers that each capsule can carry
The chocolate bars are shared among some is equal to the total number of capsules on the
students. Each student gets only one type of Singapore Flyer.
chocolate bar and every student gets the same
number of chocolate bars. If each student gets the (a) Find the prime factorisation of 784 and
greatest number of chocolate bars, express your answer in index notation.

(a) how many chocolate bars does each student (b) Hence, find the total number of capsules on
get? the Singapore Flyer.

(b) how many students will get chocolate bars
with peanuts?

15. A rectangular board measures 630 cm by 396 cm.
It is divided into small squares of equal size.

(a) (i) Find the largest possible length of the
side of a square.

(ii) Find the least number of squares.
(b) (i) Find the second largest possible length

of the side of a square.
(ii) Find the number of squares in this

case.

Write in Your Journal

1. Write down the steps involved in using prime factorisation to find the HCF and LCM of two
numbers. You may use examples to help you explain.


2. Give one example of how HCF and LCM are applied in real life.

21