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CONTENTS

INTEGERS

-9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9

EVEN-ODD NUMBERS

INTEGERS

CONSECUTIVE
NUMBERS

Consecutive
Positive Integers
are integers that
follow each other in
order:
1, 2, 3, 4, 5, …

CONSECUTIVE
INTEGERS

are even integers that
follow each other in
order:
2, 4, 6, 8, 10, …

EVEN NUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40

CONSECUTIVE EVEN
INTEGERS

Consecutive Odd
Integers
are odd integers that
follow each other in
order:
1, 3, 5, 7, …

ODD NUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40

CONSECUTIVE ODD
INTEGERS

even eveneven odd oddodd
odd oddodd even eveneven
…, n – 2, n – 1, n , n + 1, n + 2, n + 3, …
EVEN/ODD NUMBERS

even even even
odd odd even
even odd odd
odd even odd
 
 
 
 
EVEN / ODD NUMBERS

EVEN / ODD NUMBERS

even even even
odd even even
even odd even
odd odd odd
 
 
 
 
EVEN / ODD NUMBERS

EVEN / ODD NUMBERS

Consecutive Prime
Numbers
are prime numbers
that follow each other
in order:
2, 3, 5, 7, 11, …

CONSECUTIVE
NUMBERS

COUNTING
INTEGERS

COUNTING CONSEC.
INTEGERS

Counting Consecutive Integers
12, 13, 14, 15, 16, 17, 18, 19, 20
20 – 12 + 1 = 9

COUNTING CONSEC.
EVEN/ODD INTEGERS
If the result is an integer number,
that is the answer.

Counting Consecutive
Even/Odd Integers
12, 13, 14, 15, 16, 17, 18, 19
19 – 12 + 1 = 8
8 / 2 = 4

Counting Consecutive
Even/Odd Integers
11, 12, 13, 14, 15, 16, 17, 18
18 – 11 + 1 = 8
8 / 2 = 4

Subtract the smallest number
from the largest number and
add 1, divide by 2.
COUNTING CONSEC.
EVEN/ODD INTEGERS
If the result is not an integer number,
see how the series starts and ends.

Counting Consecutive
Even/Odd Integers
11, 12, 13, 14, 15, 16, 17, 18, 19
19 – 11 + 1 = 9
9 / 2 = 4.5

Counting Consecutive
Even/Odd Integers
10, 11, 12, 13, 14, 15, 16, 17, 18
18 – 10 + 1 = 9
9 / 2 = 4.5

CONSECUTIVE
NUMBERS

DIVISIBILITY

FACTOR / DIVISOR

FACTOR / DIVISIOR

a number that can be
divided by another
number without a
remainder.

MULTIPLE / DIVISIBLE

MULTIPLE / DIVISIBLE

Multiples of 2

Multiples of 3

DIVISIBILITY FACTS
1 is a factor/divisor of every integer.
0 is a multiple of every integer.
The factors of an integer include
positive and negative integers.

Factors of 4
Factors of 12

Prime Numbers
are natural numbers
that has no positive
divisors other than 1
and itself.

PRIMENUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 …

If is prime
number then, it
won’t have any
factor such that
.

2 1, 2
3 1, 3
5 1, 5
7 1, 7
11 1, 11
13 1, 13
17 1, 17
19 1, 19

Current Largest Prime
257,885,161 – 1
17,425,170 digits long
Jan 25, 2013,
University of Central Missouri

Composite
Number
a natural number
greater than 1 that is
not a prime number.

COMPOSITE
NUMBERS
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 46 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 …

If k is composite
number then, it
will have at least
one factor p such
that 1 < p < k.

4 1, 2, 4
6 1, 2, 3,6
8 1, 2, 4, 8
9 1, 3, 9
12 1, 2, 3, 4, 6, 12
14 1, 2, 7, 14
15 1, 3, 5, 15
16 1, 2, 4, 8, 16

1 1 11 1, 11 21 1, 3, 7, 21
2 1, 2 12 1, 2, 3, 4, 6, 12 22 1, 2, 11, 22
3 1, 3 13 1, 13 23 1, 23
4 1, 2, 4 14 1, 2, 7, 14 24 1, 2, 3, 4, 6, 8, 12, 24
5 1, 5 15 1, 3, 5, 15 25 1, 5, 25
6 1, 2, 3, 6 16 1, 2, 4, 8, 16 26 1, 2, 13, 26
7 1, 7 17 1, 17 27 1, 3, 9, 27
8 1, 2, 4, 8 18 1, 2, 3, 6, 9, 18 28 1, 2, 4, 7, 14, 18
9 1, 3, 9 19 1, 19 29 1, 29
10 1, 2, 5, 10 20 1, 2, 4, 5, 10, 20 30 1, 2, 3, 5, 6, 10, 15, 30

Prime
Factorization
is the decomposition
of a composite number
into prime factors,

which when multiplied
together equal the
original integer.

PRIME FACTORIZATION

PRIME FACTORIZATION
Eighter way, the result is
2  2  3  5 = 60 or 22  3  5 = 60
60
6 10
2 3 2 5
60
2 30
3
2 15
5

NUMBER OF DIVISORS
60 = 22  31  51 = 60
1 x 60
2 x 30
3 x 20
4 x 15
5 x 12
6 x 10 3 x 2 x 2 = 12
• Take all the exponents from the
prime factorization and add 1 to
each of them.
• Multiply the modified exponents
together.

of two integers is the
largest positive integer that
divides the numbers
without a remainder.

GCF

GCF
Prime factors of :
18 = 2 × 3 × 3
Prime factors of :
24 = 2 × 2 × 2 × 3
There is one 2 and one 3 in common.
The GCF of 18 and 24 is 2 × 3 = 6

GCF (GCD)
36
4 9
2 2 3 3
54
6 9
2 3 3 3
Shared Factors: 2, 3, 3
Multiply (GCF): 2  3  3 = 18
Find the GCF of 36 and 54:

The Least Common
Multiple (LCM)
of two integers or more
integers, is the smallest
positive integer that is
divisible by all the
numbers.

LCM

LCM

LCM

Factors Multiples
1 2 3 4 6 12 12 24 36 48 60 72 84 96 108 …
1 2 3 6 9 18 18 36 54 72 90 108 126 …
GCF = 6 LCM = 36
GCF and LCM

The remainder
is the amount "left
over" after performing
the division of two
integers which do not
divide evenly.

REMAINDER
1
- 6
3
2 7divisor
remainder
dividend
quotient
7 = 2∙3 + 1
dividend = divisor∙quotient + remainder

The remainder r when
n is divided by a
nonzero integer d is
zero if and only if n is
a multiple of d.

Dividing by 4

Divisible by
means that when you
divide one number by
another the result is a
whole number.

DIVISIBILITY BY 2
2, 40, 258, 1020
Last digit is even

DIVISIBILITY BY 3
69  6+9 = 15
504  5+0+4 = 9
1938  1+9+3+8 = 21
Sum of digits is a multiple of 3

DIVISIBILITY BY 4
512, 720, 1424, 1620
Last two digits are multiple of 4

DIVISIBILITY BY 5
25, 50, 560, 1005
Last digit is 5 or 0

DIVISIBILITY BY 6
72  7+2 = 9
1200  1+2+0+0 = 3
1860  1+8+6+0 = 15
Sum of the digits is multiple
of 3 and the last digit is even

DIVISIBILITY BY 7
3101  310 – 2 = 308
308  30 – 16 = 14
Take the last digit off the
number, double it and
subtract the doubled number
from the remaining number

DIVISIBILITY BY 9
729  7+2+9 = 18
810  8+1+0 = 9
9918  9+9+1+8 = 27
Sum of digits is a multiple of 9

DIVISIBILITY BY 10
30, 70, 100, 250, 560
Last digit is 0

Divisibility Rules
A number is divisible by … Divisible
Not
Divisible
2 If the last digit is even 3,728 357
3 If the sum of the digits is a multiple of 3 120 155
4 If the last two digits form a number divisible by 4 144 142
5 If the last digit is 0 or 5 150 123
6 If the number is divisible by both 2 and 3 48 20
9 If the sum of the digits is divisible by 9 729 811
10 If the last digit is 0 50 53

DIVISIBILITY

FRACTIONS

EQUIVALENT
FRACTIONS

NAMING FRACTIONS

FRACTIONS
It is useful to think of a fraction
bar as a symbol for division.
The denominator of a fraction
can’t be equal to zero.

SIGNS IN A FRACTION
numerator
fraction
denominator
Any two of the
three signs of a
fraction may be
changed without
altering the value
of the fraction.

SIGNS IN A FRACTION
2 2 2 2
5 5 5 5
 
    
 
2 2 2 2
5 5 5 5
 
    
 

COMPARING FRACTIONS
Same Denominator

COMPARING FRACTIONS
Same Numerator
1
3
1
4
1
5
1
6

5 4
?
8 7
Cross- multiplication
COMPARING
FRACTIONS
5 7?4 8
35 32
5 4
8 7
 



2 7
15 15

Make sure the
denominators
are the same.
Add the
numerators, put
the answer over
the denominator.
Simplify the
fraction.
ADDING FRACTIONS
9
15

3
5


Make sure the
denominators
are the same.
Subtract the
numerators. Put the
answer over the
same denominator.
Simplify the
fraction.
SUBTRACTING
FRACTIONS
2 9 1
15 10 5
 
25
30

4 27 6
30
 

4 27 6
30 30 30
  
5
6


5 6
7 4

3 3 7 21
7
5 5 1 5
   
Multiply the
numerators.
Multiply the
denominators.
Simplify
the
fraction.
MULTIPLYING
FRACTIONS
30
28

15
14


1 3
2 5

Turn the second
fraction upside-down
(this is now a
reciprocal).
Multiply the first
fraction by that
reciprocal.
Simplify
the
fraction.
DIVIDING FRACTIONS
1 5
2 3
 
1 5
2 3



5
6


2
2
3
 
 
 
Distribute the
exponent into the
numerator as well as
into the denominator.
Evaluate the
numerator
and the
denominator.
Simplify
the
fraction.
POWER OF FRACTIONS
2
2
2
3

4
9


4
9
Distribute the root
into the
numerator as
well as into the
denominator.
Evaluate the
numerator
and the
denominator.
Simplify
the
fraction.
ROOTS OF FRACTIONS
4
9

2
3


TRICKY OPERATIONS

The reciprocal of a is .1
a

The reciprocal of 2 is .1
2

The reciprocal of is .a b 1
a b

The reciprocal of is .3
4
4
3

a
a c a d a db
c b d b c b c
d

    

COMPLEX FRACTIONS
A fraction with fractions in the
numerator or denominator.

Proper Fraction
fraction that is less than
one, with the numerator
less than the denominator.

Improper Fraction
a fraction in which the
numerator is greater
than the denominator.

Multiply the
whole number
part by the
denominator
Add the
numerator
The result is the
new numerator
(over the same
denominator)
2
5
7
MIXED NUMBER TO
IMPROPER FRACTION
5 7 2
7
 

37
7


Divide the denominator
into the numerator.
The quotient becomes
the whole number.
The remainder becomes
the new numerator.
7
2
IMPROPER FRACTION
TO MIXED NUMBER
1
3
2


MIXED NUMBERS

Part fraction whole 
3 3
100 100 75
4 4
of   
25
25
25
25
25
25
25
PART – FRACTION

Part fraction whole 
1 1
100 100 50
2 2
of   
50
50
PART – FRACTION

FRACTIONS

DECIMALS
2 decimal
places
1 decimal
place
3 decimal
places

Divide the top
of the fraction
by the bottom.
FRACTION TO DECIMAL

5
0.625
8

4
0.571428
7

FRACTION TO DECIMAL

Write down
the decimal
divided by 1.
Multiply both top
and bottom by 10
for every number
after the decimal
point.
Simplify (or
reduce) the
fraction.
DECIMAL TO FRACTION
0.75
0.75 100
1 100
 
75
100

3
4


Terminating
Decimals
When the denominator
has only factors 2, 5, a
combination of both or of
its powers.

TERMINATING
DECIMALS
1
.5
2
7
1.4
5


2
4
3 3
.06
50 2 5
3 3
.1875
16 2
 

 

Repeating
Decimals
when the denominator
has other factors than
2 and 5 or its powers.

REPEATING DECIMALS
1
0.333
3
12
0.1212
99


4
0.571428571428...
7
5
0.384615384615...
13
repeating
decimals
repeating
decimals


1
0.333
3
12
0.1212
99


4
0.571428571428...
7
5
0.384615384615...
13
repeating
decimals
repeating
decimals



LENGTH OF THE
CLUSTER
2nd
4th
6th
3rd
6th
9th

1 2
0.111... 0.222...
9 9
3 7
0.333... 0.777...
9 9
 
 
COMMON REP.
DECIMALS

11 12
0.1111... 0.1212...
99 99
25 83
0.2525... 0.8383...
99 99
 
 
COMMON REP.
DECIMALS

127 215
0.127127... 0.215215...
999 999
853 615
0.853853... 0.615615...
999 999
 
 
COMMON REP.
DECIMALS

OPERATIONS
WITH
DECIMALS

ADDING DECIMALS
Line up decimal
points.
132.7
96.543
229.243


SUBTRACTING
DECIMALS
Line up decimal
points.
132.7
96.543
36.157


It is not necessary
to align the
decimal points.
Add the number of digits to the
right of the decimal points in the
decimals being multiplied.
MULTIPLYING
DECIMALS

125.3
1.2
2506
1253
150.36

MULTIPLYING
DECIMALS
12.53
1.2
2506
1253
15.036


Move the decimal
point in the divisor to
the right until the
divisor becomes an
integer.
Move the
decimal point in
the dividend the
same number of
places.
Proceed
with the
division.
DIVIDING DECIMALS

1.6 128.32
80.2
160 12832
1280
320
320
0


DIVIDING DECIMALS
1.6 12.832
8.02
1600 12832
12800
3200
3200
0



DECIMALS

PERCENTS

Percents:
a percentage is a
number or ratio
expressed as a
fraction of 100.

PERCENTS
Percent means
hundredths or
number out of
100.
1%
2%
20%

PERCENTS
Percent means
hundredths or
number out of
100.
%
100
1
1%
100
2
2%
100
20
20%
100
n
n 




PERCENT EQUIVALENTS
1
4
1
2
3
4
25%
0.25
50%
0.50
75%
0.75

PERCENT EQUIVALENTS
1
6
1
3
2
3
16.6%
0.1666
33.33%
0.333
66.66%
0.666

PERCENT EQUIVALENTS
1
10
1
5
1
2
10%
0.1
20%
0.20
50%
0.5

100
percent
Part whole 
PERCENTS FORMULA

PERCENTS FORMULA
12
25
100
x  

45 9
100
x
 

60
15
100
x 

25
40 160
100
 

Percent increase
Is the ratio of the
increase of two
numbers divided by
the original number
multiplyied by 100.

100%
increase
Percent increase
original whole
 

100%
(100 + n)%
n %
100%
increase
Percent increase
original whole
 

PERCENT INCREASE
The price of a tour goes up from
$80 to $100. What is the percent
increase?
20
100% 25%
80
Percent increase   

PERCENT INCREASE

The price of a tour goes up from $80 to
$100. What is the percent increase?
20
100% 25%
80
Percent increase   

100%
decrease
Percent decrease
original whole
 

(100 – n) %
100 %
n %
100%
decrease
Percent decrease
original whole
 

PERCENT DECREASE
The price of a tour goes down from
$100 to $80. What is the percent
decrease?
20
100% 20%
100
Percent decrease   

PERCENT DECREASE

COMBINED PERCENT
INCREASE
A price went up 10% one year, and the
new price went up 20% the next year.
What is the combined percent
increase?
 32% increase
110 120
100 132
100 100
  

COMBINED PERCENT
DECREASE
A price went down 10% one year, and
the new price went down 20% the next
year. What is the combined percent
decrease?
 28% decrease
90 80
100 72
100 100
  

COMBINED PERCENT
INC/DEC
A price went down 20% one year, and
the new price went up 10% the next
year. What is the combined percent
decrease?
 12% decrease
80 110
100 88
100 100
  

COMBINED PERCENT
INC/DEC
INICIAL
AMMOUNT
%
INCREASE /
DECREASE
PARTIAL
RESULT
%
INCREASE /
DECREASE
FINAL
RESULT
100 + 10% 110 -10% 99
100 - 10% 90 + 10% 99
100 + 20% 120 - 20% 96
100 - 20% 80 + 20% 96
100 + 50% 150 - 50% 75
100 - 50% 50 + 50% 75

100 100
100 100
100 100
n n 
  
100 100
100 100
100 100
n n 
  
COMBINED PERCENT
INC/DEC

INTEREST

Interest
is a fee paid by a
borrower of assets to the
owner as a form of
compensation for the use
of the assets.

INTEREST

 1
I P r n
F P I
F P rn
  
 
 
Simple interest (I) is determined
by multiplying the interest rate (r)
by the principal (P) by the
number of periods (n).
SIMPLE INTEREST

SIMPLE INTEREST
 1
I P r n
F P I
F P rn
  
 
 

SIMPLE INTEREST
Carine deposits $ 1,000 into a special bank account
which pays a simple annual interest rate of 5% for 3
years. How much will be in her account at the end of the
investment term?
P = 1,000
r = 5% = 0.05
n = 3
 
 
1
1,000 1 0.05 3
1,150
F P rn
F
F
 
   


5
5% 1,000 1,000 50
100
of   
SIMPLE INTEREST
5
5% 1,000 1,000 50
100
of   
5
5% 1,000 1,000 50
100
of   
Simple Interest on 1,000.00 after:

SIMPLE INTEREST

Interest (I) calculated on the initial
principal (P) and also on the
accumulated interest of previous
periods of a deposit or loan.
 1
n
F P r
I F P
 
 
COMPOUND INTEREST

COMPOUND INTEREST
 1
n
F P r
I F P
 
 

Annual rate = 12%, compounded:
0 12 months6
6% 6%
COMPOUND INTEREST

Principal = $ 100, Annual rate = 12%,
Time = 1 year, compounded:
COMPOUND INTEREST
 
1
100 1 0.12F  
 
2
100 1 0.06F  
 
4
100 1 0.03F  
 
12
100 1 0.01F  

COMPOUND INTEREST
Carine deposits $ 1,000 into a special bank account
which pays a compound annual interest rate of 5% for 3
years. How much will be in her account at the end of the
investment term?
P = 1,000
r = 5% = 0.05
n = 3
 
 
3
1
1,000 1 0.05
1,157.625
n
F P r
F
F
 
 


COMPOUND INTEREST
5
5% 1,000.00 1,000.00 50.00
100
of   
5
5% 1,050.00 1,050.00 52.50
100
of   
5
5% 1,102.50 1,102.50 55.13
100
of   
Compound Interest on 1,000.00 after:

COMPOUND INTEREST

SIMPLE INTEREST vs
COMPOUND INTEREST
Principal
Compound
Interest
Simple
Interest
Compound
Interest
Simple
Interest
Compound
Interest
SimpleInterest
1,050.00
1,100.00
1,150.00
P = 1,000.00
r = 5% = 0.05
n = 3 years

PERCENTS

SUMMARY

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