number patterns and sequences slide (ika) final!!

CHAPTER 2:NUMBER PATTERNS

AND SEQUENCESBy: NURSYAFIKA BINTI MOHD SHAHADAN

Number Patterns and Sequences (Corak/Pola Nombor dan Urutannya)

Odd and Even Numbers (Nombor Ganjil dan Genap)

Prime Number (Nombor Perdana)

Factors (Faktor) Prime Factors (Faktor Perdana) Multiples (Gandaan) Multiples

and Lowest Common Multiple, LCM (Gandaan Sepunya dan Gandaan Sepunya Terkecil, GSTK)

Common Factors and Highest Common Factor, HCF (Faktor Sepunya dan Faktor Sepunya Terbesar, FSTB)

Number Patterns and Sequences (Corak/Pola Nombor dan Urutannya)

NUMBER PATTERNS

AND SEQUENCES

(CORAK/POLA NOMBOR

DAN URUTANNYA)

sequenceThe numbers are arranged in a specific pattern known as the order.

Pattern numberPattern sequence number can be determined by add, subtract, multiply or divide 'previous numbers in the sequence', with the number / certain numbers.

Fibonacci Sequence

Mathematicians have studied the pattern for centuries. Patterns of numbers 1, 1, 2, 3, 5, 8, ... are called the Fibonacci sequence.

This sequence begins with 1, 1 and each after the term of the latter, obtained by adding the two terms before appearing in the thread.

Describe patterns / pattern sequence of numbers

Illustrate the trend of each following sequence of numbers:

Odd and Even Numbers (Nombor Ganjil

dan Genap)

Odd numberNumbers 1, 3, 5, 7, ... known as an odd number.

even numberNumber 2, 4, 6, 8, ... known as an even number.

Identify and explain the odd and even numbers.Example 1:Identify and specify all odd and even numbers appearing in the sequence of numbers 16, 21, 26, 31, ..., 71.

Answer:Odd numbers are 21, 31, 41, 51, 61 and 71. These numbers form a sequence of numbers obtained by adding 10 to the previous number.

Even numbers 16, 26, 36, 46, 56 and 66. These numbers form a sequence of numbers obtained by adding 10 to the previous number.

Example 2:

3 + 5 = 87 + 13 = 2019 + 25 = 24

General statement about the amount / sum of two odd numbers:

Odd number + Odd Number = Even Number* The sum of two odd numbers is an even number.

Prime Number

(Nombor Perdana)

Prime numberwhole number that can only be divided by itself and the number 1 (the number itself and number 1). Thus, a prime number has only two divisor (the number itself and the number 1).

The smallest prime number is a number 2, the only even number that is prime.

Primes less than 50 are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43 and 47.

Number 1 is NOT a prime number (NOT a prime number).

Determine whether a given number is a prime numberexample:Determine whether the following numbers are prime numbers.

13Answer:13 ÷ 1 = 1313 ÷ 13 = 113 can only be divided by 1 and 13 → (2 dividers / divisors)Thus, 13 is a prime number.

51Answer:51 ÷ 1 = 5151 ÷ 3 = 1751 ÷ 17 = 351 ÷ 51 = 5151 is divisible by 1, 3, 17 and 51 → (4 divider / divisors)Therefore, 51 is not a prime number.

Factors (Faktor)

Factora whole number that is given is that the number is a multiple of the number accurately.

1 and itself is a factor of any given number.

Find all the factors of:18Answer:18 ÷ 1 = 1818 ÷ 2 = 918 ÷ 3 = 618 ÷ 6 = 318 ÷ 9 = 218 ÷ 18 = 1

18 can be divided by 1, 2, 3, 6, 9 and 18. Therefore, the factors of 18 are 1, 2, 3, 6, 9 and 18.

50Answer:50 ÷ 1 = 5050 ÷ 2 = 2550 ÷ 5 = 1050 ÷ 10 = 550 ÷ 25 = 250 ÷ 50 = 1

50 can be divided by 1, 2, 5, 10, 25 and 50. Therefore, the factors of 50 are 1, 2, 5, 10, 25 and 50.

To determine whether a number is a whole number of factors to another.

example:Determine whether;

7 is a factor of 119.Answer:119 ÷ 7 = 17119 can be divided exactly by 7. Thus, 7 is a factor of 119.

4 is a factor of 599.Answer:599 can not be divided exactly by 4. Therefore, 4 is not a factor to 599.

Prime Factors

(Faktor Perdana)

Prime factorprime factors for a whole number is, the prime factors of the number.

Identify the prime factors of a list of factors.

example:Given 1, 2, 4, 7, 8, 14 and 56 are the factors of 56. Identify all the prime factors of 56.

Answer:Among the factors of 56, 2 and 7 is a prime number. Therefore, the prime factors of 56 are 2 and 7.

Finding the prime factors of whole numbers.example:Get a prime factor numbers: 100

Method 1

List all the factors of 100.Factor of 100 are 1, 2, 4, 5, 10, 20, 25, 50 and 100. Among all these factors, 2 and 5 are prime numbers. Therefore, the prime factors of 100 are 2 and 5.

Method 2

Using the algorithm (repeated division by prime factors).

Therefore, the prime factors of 100 are 2 and 5.

Method 3

Using a diagram trees (factor tree diagram).

From the diagram, the prime factors of 100 are 2 and 5.

Multiples (Gandaan)

MultiplesMultiples of a whole number is the product of the number of any other whole number, except zero (zero).

Multiple of the number n is in the form nk, where k = 1, 2, 3, 4, ...For example, multiple 3 = 3 x 1, 3 x 2, 3 x 3, 3 x 4, ..

Whole number is divisible by another

number if the balance is zero.

DIVIDER METHOD EXAMPLE

2 The last digit (unit value of) a number is 0, 2, 4, 6 or 8.

90, 152, 3 866, 5 478

3 The sum of all the digits is divisible by 3. 249(2 + 4 + 9) ÷ 3= 15 ÷ 3 = 5

4 Number formed by the last two digits of the number is divisible by 4 or is zero.

7 21616 ÷ 4 = 4

5 The last digit (unit value of the) number is 0 or 5. 480, 3 6256 The number is divisible by 2 and 3. 738

(7 + 3 + 8) ÷ 3= 18 ÷ 3 = 6

8 Number formed by the last three digits of the number is divisible by 8.

53 288

9 The sum of all the digits is divisible by 9. 4 302(4 + 3 + 0 + 2) ÷ 9= 9 ÷ 9 = 1

10 The last digit (unit value of the) number is 0. 560, 29 710

Divisibility Test

2Answer:= 2 x 1, 2 x 2, 2 x 3, 2 x 4, 2 x 5= 2, 4, 6, 8, 10

5Answer:= 5 x 1, 5 x 2, 5 x 3, 5 x 4, 5 x 5= 5, 10, 15, 20, 25

9Answer:= 9 x 1, 9 x 2, 9 x 3, 9 x 4, 9 x 5= 9, 18, 27, 36, 45

15Answer: = 15 x 1, 15 x 2, 15 x 3, 15 x 4, 15 x 5= 15, 30, 45, 60, 75

* The gain of the given numbers also form a sequence of numbers.

Lists multiples of whole numbersExample 1:List the first five multiples;

To determine whether a number is divisible by another number.Example 2:Determine whether 63 is divisible by;

7Answer: 63 ÷ 7 = 9 ← 63 = 7 x 9Thus, 63 is a multiple of 7.

8Answer: 63 ÷ 8 = 7, the remaining 7Therefore, 63 is not a multiple of 8.

** If the number is divisible by the number m, then n is a multiple of m.

Example 3:Use divisibility test to determine whether it is a gain of 639 234;

4Answer:Last two digits of 639 234, which is 34, can not be divided by 4.Thus, 639 234 not a multiple of 4.

9Answer:6 + 3 + 9 + 2 + 3 + 4 = 27The sum of all digits of 639234 is divisible by 9.Thus, 639 234 is a multiple of 9.

Multiples and Lowest Common Multiple, LCM (Gandaan Sepunya

dan Gandaan Sepunya Terkecil

, GSTK)

Common multiplesCommon Multiples given set of whole numbers is a multiple of each number in the set.

Lowest common multiples (LCM)Lowest Common Multiple (LCM) of some given number is the smallest common multiple of the numbers.

* The concept of 'multiple' and 'factor' is a contradiction.for example;30 is a multiple of 1, 2, 3, 5, 6, 10, 15 and 30.Meanwhile, 1, 2, 3, 5, 6, 10, 15 and are factors of 30.

Finding common multiples of two or three whole numbers.Example 1:Get common multiple of;3 and 4.Answer:Multiples of 3: 3, 6, 9, 12, 15, 18, 21, 24, ...Multiples of 4: 4, 8, 12, 16, 20, 24, 28, ...Common multiples of 3 and 4 are 12, 24, 36, ...

2, 3 and 6.Answer:Multiples of 2: 2, 4, 6, 8, 10, 12, 14, 16, 18, ...Multiples of 3: 3, 6, 9, 12, 15, 18, 21, ...Multiples of 6: 6, 12, 18, 24, 36, ...Common multiples of 2, 3 and 6 are 6, 12, 18, ...

** List some common multiples of whole numbers is also a sequence of numbers.

To determine whether a number is a common multiple of two or three whole numbers that are given.Example 2:Determine whether;84 is a common multiple of 5 and 7.Answer:84 ÷ 5 = 16 remainder 484 ÷ 7 = 1284 can not be divisible by 5.Therefore, 84 is not a Common Multiples of 5 and 7.

432 is a common multiple of 6, 8 and 9.Answer:432 ÷ 6 = 72432 ÷ 8 = 54432 ÷ 9 = 48432 is divisible by 6, 8 and 9.Therefore, 432 is Common Multiples of 6, 8 and 9.

Determine LCM of two whole numbers.

Example 3:Find the Least Common Multiples for;9 and 12Answer:

Method 1: Factoring Prime (Prime Factorisation)

LCM for 9 and 12 = 3 x 3 x 2 x 2 = 36

Method 2: Use the algorithm (repeated division by prime factors)

LCM for 9 and 12 = 3 x 3 x 2 x 2 = 36

Example 4:Determine LCM for;

6, 15 and 18.Answer:

LCM 6, 15 and 18 = 2 x 3 x 3 x 5 = 90

14, 28 and 49.Answer:

LCM for 14, 28, 49 = 7 x 2 x 2 x 7 = 196

Common Factors and Highest Common Factor, HCF (Faktor Sepunya dan

Faktor Sepunya Terbesar, FSTB)

Common factorsCommon Factors some whole number is a number that is a factor of each of those numbers.

Highest common factor (hcf)Highest Common Factor (HCF) number assigned number is the largest number that is a factor of each of those numbers.

Finding a common factor of two or three whole numbers.Example 1:Find the common factors;18 and 54.Answer:Factors of 18: 1, 2, 3, 6, 9, 18Factor of 54: 1, 2, 3, 6, 9, 18, 27, 54Common factors of 18 and 54 are 1, 2, 3, 6, 9 and 18.

9, 15 and 21.Answer:Factor of 9: 1, 3, 9Factor of 15: 1, 3, 5, 15Factor of 21: 1, 3, 7, 21Common factor of 9, 15 and 21 are 1 and 3.

To determine whether a number is a factor common to two or three given numbers.Example 2:Determine whether;12 is a common factor of 84 and 156.Answer:84 ÷ 12 = 7156 ÷ 12 = 13Thus, 12 is a common factor of 84 and 156.

4 is a common factor for 32, 70 and 112.Answer:32 ÷ 4 = 870 ÷ 4 = 17 remainder 2112 ÷ 4 = 28Therefore, 4 is not a common factor of 32, 70 and 112.

Determine the Greatest Common Factor of two whole numbers.

Example 3:Get the highest common factor of;28 and 32.Answer:

Method 1: List all the factors for each number. Factors of 28: 1, 2, 4, 7, 14, 28 Factor of 32: 1, 2, 4, 8, 16, 32 Therefore, the HCF of 28 and 32 is 4.

Method 2: Use the algorithm (repeated division by a common factor).

Greatest common factor of 28 and 32 is = 2 x 2 = 4.

Example 4:Get the highest common factor (HCF) for;

40, 48 and 56.Answer:

* Distribution discontinued as 5, 6 and 7 have no common factors other than 1.

Thus, the HCF for 40, 48 and 56= 2 x 2 x 2= 8

70, 84 and 126.Answer:

* Distribution discontinued as 5, 6 and 9 have no common factors other than 1.

Thus, the HCF for 70, 84 and 126= 2 x 7= 14

19, 38, p, q, 304, ….. is a number sequence.Evaluate p + q.a. 76b. 152c. 228d. 310

Number Patterns and Sequences (Corak/Pola Nombor dan Urutannya)

State whether each of the following numbers is an even number. 

(a) 17                    (c) 44

(b) 60                    (d) 95

Odd and Even Numbers (Nombor Ganjil dan Genap)

Prime Number (Nombor Perdana)

Find the sum of all the prime numbers which are less than 10.

Factors (Faktor)

Determine whether 9 is a factor of the following numbers.

(a) 144          (b) 322

Prime Factors (Faktor Perdana)

Determine which of the following are prime factors of 84.

(a) 3           (b) 21

Multiples (Gandaan)

Determine whether 592 is a multiple of

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

Multiples and Lowest Common Multiple, LCM (Gandaan Sepunya dan Gandaan Sepunya Terkecil, GSTK)Which of the following numbers is the lowest common multiple (LCM) of 12, 32 and 48?a. 96b. 192c. 284d. 384

Common Factors and Highest Common Factor, HCF (Faktor Sepunya dan Faktor Sepunya Terbesar, FSTB)Find the highest common factor (HCF) of 16, 24 and 32.a. 2b. 4c. 8d. 16