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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