numbers patterns and sequences

8/17/2019 Numbers Patterns and Sequences

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NUMBERS PATTERNS AND SEQUENCES

NUMBER PATTERNS AND SEQUENCES


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A) Pattern of a Number Sequence

1. A sequence in a ist numbers fo o!in" a certain #attern .

$. Eac% number is ca e& a term of t%e sequence.

'or e(am# e *

+or,e& E(am# e

Describe t%e #attern of eac% of t%e fo o!in" sequences

-a) $ / $0/ 1 / 1$/ 2/..

-b) 1/ 3/ 4/ $5/ 21/...

-c) 0/ 1/ 1/ $/ 3/ 6/ 2/..

So ution


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B) Com# etin" a Sequence

To com# ete a sequence/ first i&entif7 t%e number #attern. +or,e& e(am# e

So ution


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C) Constructin" a Sequence

A sequencecan be forme& if t%e #attern is "i8en.

+or,e& E(am# e

Construct a fi8e * number sequence for eac% of

t%e fo o!in".

-a) Startin" !it% 5/ a&& to t%e number before it.

-b) Startin" !it% $/ mu ti# 7 t%e number before it b7 .

-c) Startin" !it% $60/ &i8i&e t%e number before it b7 6.

So ution

9DD NUMBERS AND E:EN NUMBERS

A) Determinin" 9&& Numbers an& E8en Numbers


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1. E8en numbers are numbers t%at can be &i8i&e& b7 $ e(act 7 .

'or e(am# e *

$/ / / 2/ 10/....

$. 9&& numbers are numbers t%at cannot be &i8i&e& b7 $ e(act 7 .

'or e(am# e *

1/ 3/ 6/ 5/ 4/...

3. 0 is neit%er o&& nor e8en.

+or,e& e(am# e

State !%et%er eac% of t%e fo o!in" numbers is an

e8en number.

-a) 15 -c)

-b) 0 -&) 46

So ution

-a) 9&& number -c) E8en number

-c) E8en number -&) 9&& number

+or,e& e(am# e

;ist-a) t%e first fi8e e8en numbers.

-b) a t%e o&& numbers bet!een 60 an& 6.

So ution

-a) $/ / / 2/ 10

-b) 61/ 63/ 66/ 65/ 64/ 1/ 3

+or,e& e(am# e

Determine !%et%er t%e ans!er for eac% of t%e

fo o!in" is an o&& or e8en number.

-a) 6 < 5

-b) 5 *

-c) 2 ( 2


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So ution

PR=ME NUMBERS

A) Determinin" Prime Numbers

1. A #rime number is a !%o e number t%at can on 7

be &i8i&e& b7 itse f an& 1 .

$. T%us/ $/ 3/ 6/ 5/ 11/ 13/ 15/... are #rime numbers.

3. T%e first #rime number is $.

. To &etermine !%et%er a "i8en number is a #rime number/ carr7 out t%e fo o!in" ste#s.

Ste#s 1 Di8i&e t%e "i8en number b7 #rime numbers !%ose squares are ess t%an it.

Ste#s $ =f t%e "i8en number cannot be &i8i&e& e(act 7 b7 a t%e #rime numbers/ t%an it is a#rime number.

+or,e& e(am# e

Determine !%et%er eac% of t%e fo o!in" is a #rime numbers or not.

-a) 6 -b) 53

So ution


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+or,e& e(am# e

;ist a t%e #rime numbers bet!een 30 an& 0.

So ution

+rite &o!n a t%e o&& numbers bet!een 30

an& 0 an& t%en cross out a t%e numbers &i8isib e

b7 3/ 6/ 5.

31/ 33 / 36 / 35/ 34 / 1/ 3/ 6/ 5/ 4 / 61 / 63/ 66 /

65 / 64

T%erefore/ t%e #rime numbers bet!een 30 an&

0 are 31/ 35/ 1/ 3/ 5/ 63 an& 64.

+or,e& e(am# e

'in& t%e sum of a t%e #rime numbers !%ic% are ess t%an 10.

So utionPrime numbers !%ic% are ess t%an 10 are $/ 3/ 6 an& 5.


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T%erefore/ its sum > $< 3


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-b) 'irst fi8e mu ti# es of 1 0

> 1 0 ( 1/ 1 0 ( $/ 1 0 ( 3/ 1 0 ( / 1 0 ( 6

> 1 0/ $20/ $0/ 6 0/ 500

B) Determinin" Mu ti# es

1. =f !%o e number p is &i8isib e b7 !%o e number q ,

t%en p is t%e mu ti# e of q .

'or e(am# e *

16/ $0/ an& $6 are a &i8isib e b7 6. T%erefore/ 16/ $0

an& $6 are a mu ti# es of 6.

$. Be o! are sim# e tests of &i8isibi it7 b7 $/ 3/ / 6/ / 5/ 2/ 4/ 10 an& 11.


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+or,e& e(am# e

Determine !%et%er 64$ is a mu ti# e of

-a) $/ -b) 3/ -c) .

So ution

-a) 64$ is an e8en number an& t%erefore &i8isib e b7 $.

T%erefore/ 64$ is a mu ti# e of $.

C%ec, 64$ @ $ > $4

-b) Sum of t%e &i"its of 64$

> 6 < 4 < $

> 1

1 @ 3 > 6 remain&er 1

T%erefore/ 64$ is not a mu ti# e of 3.


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-c) ;ast t!o &i"its of 64$ > 4$

4$ @ > $3

T%erefore/ 64$ is a mu ti# e of .

C%ec, 64$ @ > 1 2

C9MM9N MU;T=P;ES AND ;9+EST C9MM9N MU;T=P;E - ;CM )

A) 'in&in" Common Mu ti# es

1. A common mu ti# e is a number !%ic% is a

mu ti# e of t!o or more "i8en numbers.

'or e(am# e *

30 is a mu ti# e of 3. 30 is a so a mu ti# e of 6.

T%erefore/ 30 is ,no!n as t%e common mu ti# e

of 3 an& 6.

$. Common mu ti# es of t!o or more "i8en numbers

can be foun& b7 istin" t%e mu ti# es of t%ese numbers.

+or,e& e(am# e

'in& t%e first t%ree common mu ti# es of

-a) $ an& 3.

So ution

-a) Mu ti# es of $ > $/ / / 2/ 10/ 1$ / 1 / 1 /

12 / $0/...

Mu ti# es of 3 > 3/ / 4/ 1$ / 16/ 12 / $1/....

T%erefore/ t%e first t%ree common mu ti# es of

$ an& 3 are / 1$ an& 12.

+or,e& e(am# e

'in& t%e common mu ti# es of $/ / an& 6 t%at t%e

ess t%an 100.


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So ution

Mu ti# es of $ > $/ / / 2/ 10/ 1$/ 1 / 1 / 12/ $0/...

Mu ti# es of > / 2/ 1$/ 1 / $0/...

Mu ti# es of 6 > 6/ 10/ 16/ $0/...

T%erefore/ common mu ti# es of $/ an& 6 t%an are ess t%an 100.

> - $0 ( 1 )/ - $0 ( $ )/ - $0 ( 3 )/ - $0 ( )

> $0/ 0/ 0/ 20

B) Determinin" t%e Common Mu ti# es

A number is a common mu ti# e of a "rou# of "i8en

numbers if it is &i8isib e b7 t%ose "i8en numbers.

+or,e& e(am# e

Determine !%et%er 0 is a common mu ti# e of

-a) 3 an& /

-b) 6 an& 2/

-c) / 1$ an& 16.

So ution

-a) 0 @ 3 > $0 0 @ > 16

0 is &i8isib e b7 3 an& .

T%erefore/ 0 is a common mu ti# e of 3 an& .

-b) 0 @ 6 > 1$ 0 @ 2 > 5 remain&er .

0 is &i8isib e b7 6 but not 2.

T%erefore/ 0 is not a common mu ti# e of 6 an& 2.

-c) 0 @ > 10 0 @ 1$ > 6 0 @ 16 >

0 is &i8isib e b7 / 1$ an& 16.

T%erefore/ 0 is a common mu ti# e of / 1$ an& 16.

C) Determinin" t%e ;o!est Common Mu ti# e -;CM)

1. T%e o!est common mu ti# e of t!o or more numbers is t%e sma est common mu ti# e of

t%ese numbers.


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$. T%e ;CM can be foun& b7 t%e fo o!in" met%o&s.

Met%o& 1

;ist t%e mu ti# es of t%e "i8en set of numbers.

Met%o& $

Di8i&e t%e "i8en numbers b7 t%eir common factors

unti t%e quotients %a8e more common factor.

+or,e& e(am# e

'in& t%e ;CM of

-a) 4 an& 1$

So ution

-a) Met%o& 1

Mu ti# es of 4 > 4/ 12/ $5/ 3 /...

Mu ti# es of 1$ > 1$/ $ / 3 /...

T%erefore/ ;CM of 4 an& 1$ is 3 .

Met%o& $ - b7 usin" t%e a "orit%m )

$. 'ACT9RS

A) Determinin" t%e factors of a number

1. A factor of a "i8en number is t%e number t%at

can &i8i&e e(act 7 t%e "i8en number .

'or e(am# e *

-a) 4 is &i8isib e b7 1/ 3 an& 4.

T%erefore/ 1/ 3 an& 4 are factors of 4.

-b) 1$ is &i8isib e b7 1/ $/ 3/ / / an& 1$.

T%erefore/ 1/ $/ 3/ / an& 1$ are factors of 1$.

$.To &etermine !%et%er a number is a factor of a "i8en number/ t%e test of &i8isibi it7 cana so be use&.


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+or,e& e(am# e

Determine !%et%er 4 is a factor of t%e

fo o!in" numbers.

-a) 1 -b) 3$$

So ution

-a)

-b)

T%erefore/ 4 is not a factor of 3$$.

B) 'in&in" t%e 'actors of a Number

'actors of a "i8en number can be foun& b7

&i8i&in" t%e number b7 itse f an& ot%er sma er

numbers.

+or,e& e(am# e

'in& a t%e factor of t%e fo o!in" numbers.

-a) 16

So ution

-a)

T%erefore/ 1/ 3/ 6 an& 16 are factors of 16.


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PR=ME 'ACT9RS

A) Determine t%e Prime 'actors

T%e #rime factor of a number is t%e #rime

number t%at is a so a factor of t%at number .

'or e(am# e *

'actor of > 1/ $/ 3/ an&

Amon" t%e factors/ $ an& 3 are #rime numbers.

T%erefore/ $ an& 3 are t%e #rime factors of .

+or,e& E(am# e

Determine !%ic% of t%e fo o!in" are #rime

factors of 2 .

-a) 3 -b) $1

So ution

-a) 2 @ 3 > $2 3 is a #rime number.

T%erefore/ 3 is a #rime factor of 2 .

-b) 2 @ $1 > but $1 is not a #rime number.

T%erefore/ $1 is not a #rime factor of 2 .

B) 'in&in" Prime 'actors

T%ere are t!o !a7s to fin& t%e #rime factors of a number.

Met%o& 1

;ist a t%e factors of t%e "i8en number an& t%en #ic, out t%e #rime factor - s ).

Met%o& $

Di8i&e t%e "i8en number re#eate& 7 b7 #rime

numbers - startin" !it% t%e sma est #ossib e )

unti t%e &i8ision is com# ete&.

+or,e& e(am# e

'in& t%e #rime factors of 6$.


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T%erefore/ 3 is a common factor of 52 an& 1$0.

-b) 2 @ 3 > 1

111 @ 3 > 35

$$0 @ 3 > 53 remain&er

T%erefore/ 3 is not a common factor of 2/ 111 an& $$0.

B) 'in&in" t%e Common 'actors

To fin& t%e common factors of a "rou# of "i8en numbers/ ist a t%e factors of eac% "i8ennumber an& t%en #ic, out t%eir common factors.

+or,e& e(am# e

'in& a t%e common factors of eac% of t%e fo o!in" "rou#s a numbers.

-a) $ an& 2

So ution

-a) 'actors of $ > 1/ $/ 3/ / 5/ 1 / $1/ $

'actors of 2 > 1/ $/ 3/ / / 2/ 1$/ 1 / $ / 2

T%erefore/ t%e common factors of $ an& 2

are 1/ $/ 3/ an& .

+or,e& e(am# e

'in& t%e #ro&uct of a t%e common factors of 2/ 1$ an& 1 .

So ution

'actors of 2 > 1/ $/ / 2

'actors of 1$ > 1/ $/ 3/ / / 1$

'actors of 1 > 1/ $/ / 2/ 1

T%erefore/ t%e #ro&uct of a t%e common factors of 2/ 1$ an& 1

> 1 ( $ (

> 2

C) 'in&in" t%e i"%est Common 'actor - 'C )

1. T%e %i"%est common factor is t%e ar"est number !%ic% is a factor of t!o or morenumbers.'or e(am# e *


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T%e common factors of $ an& 3 are 1/ $/ 3/ / / an& 1$.T%erefore/ t%e %i"%est common factor - C' ) of $ an& 3 is 1$.

$. T%ere are t!o met%o&s of fin&in" t%e C' of a

"i8en "rou# of numbers.

Met%o& 1

;ist a t%e factors of eac% number an& t%en #ic, out

t%e ar"est common factor.

Met%o& $

Di8i&e t%e "i8en numbers b7 t%eir common factors an&

t%en fin& t%eir #ro&uct.

+or,e& e(am# e

'in& t%e C' of t%e fo o!in".

-a) 12 an& 6

So ution

-a) Met%o& 1

'actors of 12 > 1/ $/ 3/ / 4/ 12

'actors of 6 > 1/ 3/ 6/ 4/ 16/ 6

T%e common factors of 12 an& 6 are 1/3 an& 4.

T%erefore/ t%e C' of 12 an& 6 is 4.

Met%o& $

T%erefore/ t%e C' of 12 an& 6

> 3 3

> 4


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