MATHEMATICS - FORM 1

NUMBERS PATTERNS AND SEQUENCES

           
             

             
         

NUMBER PATTERNS AND SEQUENCES

A) Pattern of a Number Sequence

1. A sequence in a list numbers following a certain pattern.

2. Each number is called a term of the sequence.

    For example:-



Worked Example


Describe the pattern of each of the following sequences

(a) 24, 20, 16, 12, 8,..

(b) 1, 3, 9, 27, 81,...

(c) 0, 1, 1, 2, 3, 5, 8,..

Solution



B) Completing a Sequence

To complete a sequence, first identify the number pattern.
 
Worked example



Solution



C) Constructing a Sequence

A sequencecan be formed if the pattern is given.

Worked Example

Construct a five - number sequence for each of

the following.

(a) Starting with 7, add 6 to the number before it.

(b) Starting with 2, multiply the number before it by 4.

(c) Starting with 6 250, divide the number before it by 5.

Solution



ODD NUMBERS AND EVEN NUMBERS

A) Determining Odd Numbers and Even Numbers

1. Even numbers are numbers that can be divided

   by 2 exactly.

    For example:-

    2, 4, 6, 8, 10,....

2. Odd numbers are numbers that cannot be divided

   by 2 exactly.

   For example:-

   1, 3, 5, 7, 9,...

3. 0 is neither odd nor even.

Worked example

State whether each of the following numbers is an

even number.

(a) 17                    (c) 44

(b) 60                    (d) 95

Solution

(a) Odd number    (c) Even number

(c) Even number   (d) Odd number

Worked example

List

(a) the first five even numbers.

(b) all the odd numbers between 50 and 65.

Solution

(a) 2, 4, 6, 8, 10

(b) 51, 53, 55, 57, 59, 61, 63

Worked example

Determine whether the answer for each of the

following is an odd or even number.

(a) 5 + 7

(b) 7 - 4

(c) 8 x 8

Solution


PRIME NUMBERS

A) Determining Prime Numbers

1. A prime number is a whole number that can only

   be divided by itself and 1.

2. Thus, 2, 3, 5, 7, 11, 13, 17,... are prime numbers.

3. The first prime number is 2.

4. To determine whether a given number is a prime

   number, carry out the following steps.

Steps 1: Divide the given number by prime numbers

             whose squares are less than it.

Steps 2: If the given number cannot be divided exactly by

             all the prime numbers, than it is a prime number.

Worked example

Determine whether each of the following is a prime

numbers or not.

(a) 45               (b) 73

Solution



Worked example

List all the prime numbers between 30 and 60.

Solution

Write down all the odd numbers between 30

and 60 and then cross out all the numbers divisible

by 3, 5, 7.

31, 33, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55,

57, 59

Therefore, the prime numbers between 30 and

60 are 31, 37, 41, 43, 47, 53 and 59.

than 10.

Worked example

Find the sum of all the prime numbers which are

less than 10.

Solution

Prime numbers which are less than 10 are 2, 3, 5

and 7.

Therefore, its sum = 2+ 3 +5 +7 = 17

MULTIPLES

A) Listing the Multiples of a number

1. The multiple of a number is the product of that number

   and another non-zero whole number.


   For example:-

   (a) Multiples of 4

        = 4 x 1, 4 x 2, 4 x 3, 4 x 4, 4 x 5,...

        = 4, 8, 12, 16, 20,...

   (b) First five multiples of 9

     = 9 x 1, 9 x 2, 9 x 3, 9 x 4, 9 x 5

     = 9, 18, 27, 36, 45,...

2. A number is amultiple of itself and 1.

   For example:-

    7 is a multiple of  7 and 1 since

    7 = 7 x 1 and 7 = 1 x 7.

Worked example

List the first five multiples of

(a) 8               (b)140

Solution

(a) First five multiples of 8

     = 8 x 1, 8 x 2, 8 x 3, 8 x 4, 8 x 5

     = 8, 16, 24, 32, 40

(b)
First five multiples of 140

     = 140 x 1, 140 x 2, 140 x 3, 140 x 4, 140 x 5

     = 140, 280, 420, 560, 700

B) Determining Multiples

1. If whole number p is divisible by whole number q,

   then p is the multiple of q.

   For example:-

   15, 20, and 25 are all divisible by 5. Therefore, 15, 20

   and 25 are all multiples of 5.

2. Below are simple tests of divisibility by 2, 3, 4, 5, 6, 7,

    8, 9, 10 and 11.






Worked example

Determine whether 592 is a multiple of

(a) 2,          (b) 3,          (c) 4.

Solution

(a) 592 is an even number and therefore divisible by 2.

     Therefore, 592 is a multiple of 2.

     Check : 592 ÷ 2 = 296

(b) Sum of the digits of 592

     = 5 + 9 + 2

     = 16

     16 ÷ 3 = 5 remainder 1

     Therefore, 592 is not a multiple of 3.

(c) Last two digits of 592 = 92

     92 ÷ 4 = 23

     Therefore, 592 is a multiple of 4.

     Check : 592 ÷ 4 = 148

COMMON MULTIPLES AND LOWEST

COMMON MULTIPLE ( LCM )

A) Finding Common Multiples

1. A common multiple is a number which is a

   multiple of two or more given numbers.

   For example:-

   30 is a multiple of 3. 30 is also a multiple of 5.

   Therefore, 30 is known as the common multiple

   of 3 and 5.

2. Common multiples of two or more given numbers

   can be found by listing the multiples of these numbers.

Worked example

Find the first three common multiples of

(a) 2 and 3.

Solution

(a) Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16,

                             18, 20,...

     Multiples of 3 = 3, 6, 9, 12, 15, 18, 21,....

     Therefore, the first three common multiples of

     2 and 3 are 6, 12 and 18.

Worked example

Find the common multiples of 2, 4, and 5 that the

less than 100.

Solution

Multiples of 2 = 2, 4, 6, 8, 10, 12, 14, 16, 18, 20,...

Multiples of 4 = 4, 8, 12, 16, 20,...

Multiples of 5 = 5, 10, 15, 20,...

Therefore, common multiples of 2, 4 and 5 than

are less than 100.

= ( 20 x 1 ), ( 20 x 2 ), ( 20 x 3 ), ( 20 x 4 )

= 20, 40, 60, 80

B) Determining the Common Multiples

A number is a common multiple of a group of given

numbers if it is divisible by those given numbers.

Worked example

Determine whether 60 is a common multiple of

(a) 3 and 4,

(b) 5 and 8,

(c) 6, 12 and 15.

Solution

(a) 60 ÷ 3 = 20; 60 ÷ 4 = 15

     60 is divisible by 3 and 4.

    Therefore, 60 is a common multiple

    of 3 and 4.

(b) 60 ÷ 5 = 12; 60 ÷ 8 = 7 remainder 4.

    60 is divisible by 5 but not 8.

    Therefore, 60 is not a common multiple

    of 5 and 8.

(c) 60 ÷ 6 = 10; 60 ÷ 12 = 5; 60 ÷ 15 = 4

    60 is divisible by 6, 12 and 15.

   Therefore, 60 is a common multiple

   of 6, 12 and 15.

C) Determining the Lowest Common Multiple (LCM)

1. The lowest common multiple of two or more

    numbers is the smallest common multiple of

    these numbers.

2. The LCM can be found by the following methods.

    Method 1:

    List the multiples of the given set of numbers.

    Method 2:

    Divide the given numbers by their common factors

    until the quotients have more common factor.

Worked example

Find the LCM of

(a) 9 and 12

Solution

(a) Method 1:

     Multiples of 9 = 9, 18, 27, 36,...

     Multiples of 12 = 12, 24, 36,...

     Therefore, LCM of 9 and 12 is 36.

     Method 2: ( by using the algorithm )

   
2.6 FACTORS

A) Determining the factors of a number

1. A factor of a given number is the number that

   can divide exactly the given number.

   For example:-

   (a) 9 is divisible by 1, 3 and 9.

      Therefore, 1, 3 and 9 are factors of 9.

   (b) 12 is divisible by 1, 2, 3, 4, 6, and 12.

       Therefore, 1, 2, 3, 4, 6 and 12 are factors of 12.

2.To determine whether a number is a factor of a

   given number, the test of divisibility can also be

   used.

Worked example

Determine whether 9 is a factor of the

following numbers.

(a) 144          (b) 322

Solution

(a)
   
(b)
     
Therefore, 9 is not a factor of 322.

B) Finding the Factors of a Number

Factors of a given number can be found by

dividing the number by itself and other smaller

numbers.

Worked example

Find all the factor of the following numbers.

(a) 15

Solution

(a)
     
Therefore, 1, 3, 5 and 15 are factors of 15.

PRIME FACTORS

A) Determine the Prime Factors


The prime factor of a number is the prime

number that is also a factor of that number.

For example:-

Factor of 6 = 1, 2, 3, and 6

Among the factors, 2 and 3 are prime numbers.

Therefore, 2 and 3 are the prime factors of 6.

Worked Example

Determine which of the following are prime

factors of 84.

(a) 3           (b) 21

Solution

(a) 84 ÷ 3 = 28; 3 is a prime number.

     Therefore, 3 is a prime factor of 84.

(b) 84 ÷ 21 = 4 but 21 is not a prime number.

     Therefore, 21 is not a prime factor of 84.

B) Finding Prime Factors

There are two ways to find the prime factors of a number.

Method 1 :

List all the factors of the given number and then pick out the prime factor ( s ).

Method 2 :

Divide the given number repeatedly by prime

numbers ( starting with the smallest possible )

until the division is completed.

Worked example

Find the prime factors of 52.

Solution

Method 1 :

Factors of 54 = 1, 2, 3, 6, 9, 18, 27, 54

Therefore, the prime factors of 54 are

2 and 3.

Method 2 :
          
Therefore, the prime factors of 54 are 2 and 3.

2.8 COMMON FACTORS AND HIGHEST

 COMMON FACTOR ( HCF )


A) Determining the Common Factors

1. A number which is a factor of two or more numbers

   is the common factor of these numbers.

For example:-

Factors of 10 = 1, 2, 5, 10

Factor of 15 = 1, 3, 5, 15

Therefore, 1 and 5 are the common factors of 10 and 15.

2. The test of divisibility can be used determine whether

    a number is a common factor of a given group of numbers
.

Worked example

Determine whether 3 is a common factor of

(a) 78 and 120,

(b) 48, 111, 220

Solution

(a) 78 ÷ 3 = 26

     120 ÷ 3 = 40

     Therefore, 3 is a common factor of 78 and 120.

(b) 48 ÷ 3 = 16

     111 ÷ 3 = 37

     220 ÷ 3 = 73 remainder 4

     Therefore, 3 is not a common factor of 48, 111 and 220.

B) Finding the Common Factors

To find the common factors of a group of given

numbers,list all the factors of each given number

and then pick out their common factors.

Worked example

Find all the common factors of each of the following

groups a numbers.

(a) 42 and 48

Solution

(a) Factors of 42 = 1, 2, 3, 6, 7, 14, 21, 42

     Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

     Therefore, the common factors of 42 and 48

     are 1, 2, 3, and 6.

Worked example

Find the product of all the common factors of 8, 12 and 16.

Solution

Factors of 8 = 1, 2, 4, 8

Factors of 12 = 1, 2, 3, 4, 6, 12

Factors of 16 = 1, 2, 4, 8, 16

Therefore, the product of all the common

factors of 8, 12 and 16

= 1 x 2 x 4

= 8

C) Finding the Highest Common Factor ( HFC )

1. The highest common factor is the largest number

    which is a factor of two or more numbers.

    For example:-

    The common factors of 24 and 36 are 1, 2, 3, 4,

    6, and 12.

    Therefore, the highest common factor ( HCF ) of

    24 and 36 is 12.

2. There are two methods of finding the HCF of a

    given group of numbers.

    Method 1 :

    List all the factors of each number and then pick out

    the largest common factor.

    Method 2 :

    Divide the given numbers by their common factors and

    then find their product.

Worked example

Find the HCF of the following.

(a) 18 and 45

Solution

(a) Method 1 :

Factors of 18 = 1, 2, 3, 6, 9, 18

Factors of 45 = 1, 3, 5, 9, 15, 45

The common factors of 18 and 45 are 1,3 and 9.

Therefore, the HCF of 18 and 45 is 9.

Method 2 :

Therefore, the HCF of 18 and 45

= 3 * 3

= 9