Number Properties - Mgmat Guide 1

Front 1

4+5=9
(-2)+1=-1

Back 1

The sum of two integers is always an integer.

Front 2

5-5=-1
(-2)-(-3)=1

Back 2

The difference of two integers is always an integer.

Front 3

4x5=20
(-2)x3=-6

Back 3

The product of two integers is always an integer.

Front 4

8/2=4
but
2/8=1/4
and
(-8)/4=-2
but
(-8)/(-6)=4/3

Back 4

The result of dividing two integers is SOMETIMES an integer. This result is called the quotient.

Front 5

Divisibility of an integer:

Back 5

An integer is said to be divisible by another number if the integer can be divided by that number with an integer result. (i.e. no remainder)

Hint 5

21 is divisible by 3 because 21 divided by 3 yields 7 with zero remainder. 21 is not divisible by 4 because 21 divided by 4 yields 5 with a remainder of 1.

Front 6

Divisibility Rules
An integer is divisible by 2 if:

Back 6

The integer is even.

Hint 6

12 is divisible by 2 but 13 is not.

Front 7

Divisibility Rules
An integer is divisible by 3 if:

Back 7

The SUM of the integer's DIGITS is divisible by 3.

Hint 7

72 is divisible by 3 because 7+2=9 and 9 is divisible by 3. 83 is not because 11 is not divisible by 3.

Front 8

Divisibility Rules
An integer is divisible by 4 if:

Back 8

The integer is divisible by 2 TWICE, or if the LAST TWO digits are divisible by 4.

Hint 8

28 is divisible by 4 because it is divisible by 2 twice and you get an integer result (i.e. 28/2=14/2=7).
Larger numbers ex:23,456 is divisible by 4 because 56 is divisible by 4.

Front 9

Divisibility Rules
An integer is divisible by 5 if:

Back 9

The integers ends in 0 or 5.

Hint 9

75 & 80 are 77 & 83 are not.

Front 10

Divisibility Rules
An integer is divisible by 6 if:

Back 10

The integers is divisible by BOTH 2 and 3.

Hint 10

48 is divisible by 6 since it is divisible by 2 (it ends in 8 which is even) AND by 3 (4+8=12) which is divisible by 3.

Front 11

Divisibility Rules
An integer is divisible by 7 if:

Back 11

There is no rule.

Hint 11

Simplest way to check for divisibility by 7 is to do long division.

Front 12

Divisibility Rules
An integer is divisible by 8 if:

Back 12

The integer is divisible by 2 THREE TIMES, or if the LAST THREE digits are divisible by 8.

Hint 12

32 is divisible by 8 because it is divisible by 2 THREE TIMES with an integer result. (32/2=16/2=8/2=4). Larger numbers ex:23.456 is divisible by 8 because 456 is divisible by 8.

Front 13

Divisibility Rules
An integer is divisible by 9 if:

Back 13

The SUM of the integer's DIGITS is divisible by 9.

Hint 13

4,185 (4+1+8+5=18) which is divisible by 9.

Front 14

Divisibility Rules
An integer is divisible by 10 if:

Back 14

The integer ends in 0.

Hint 14

670 is and 675 is not.

Front 15

Reverse Rules Example

Back 15

A number has a ones digit equal to zero.

Hint 15

Divisible by 10.

Front 16

Factor

Back 16

A positive integer that divides evenly into an integer.

Hint 16

1, 2, 4, 8 are all the factors (also called divisors) of 8.

Front 17

Multiple

Back 17

Formed by multiplying an integer by an integer. Negative and Positive multiples exist.

Hint 17

8, 16, 24, 32 are multiples of 8 also -8,-16,-24,-32.
GMAT*does not test negative multiples directly.

Front 18

Multiple & Factor Rules #1

Back 18

Integer is always both a factor and a multiple of itself.

Hint 18

1 is always a factor of EVERY integer.
0 is always a multiple of EVERY integer.

Front 19

Fewer Factors, More Multiples

Back 19

Factors divide into an integer and are therefore less than or equal to that integer.

Hint 19

Positive multiples, multiply out from an integer and therefore greater than or equal to that integer.

Front 20

Multiple & Factor Rules #2

Back 20

An integer only has a limited number of factors.

Hint 20

An integer has an infinite number of multiples.

Front 21

Multiple & Factor Rules #3

Back 21

3 is a factor of 12

Hint 21

Same as saying 12 is a multiple of 3 or 12 is divisible by 3.

Front 22

12 is divisible by 3

Back 22

GMAT Term for Divisibility

Front 23

3 is a divisor of 12

Back 23

GMAT Term for Divisibility

Front 24

12/3 is an integer

Back 24

GMAT Term for Divisibility

Front 25

12=3n where n is an integer

Back 25

GMAT Term for Divisibility

Front 26

12 items can be shared among 3 people so that each person has the same number of items.

Back 26

GMAT Term for Divisibility

Front 27

3 divides 12

Back 27

GMAT Term for Divisibility

Front 28

12/3 yields a remainder of 0

Back 28

GMAT Term for Divisibility

Front 29

3 goes into 12 evenly

Back 29

GMAT Term for Divisibility

Front 30

3 is a factor of 12

Back 30

GMAT Term for Divisibility

Front 31

If you add two multiples of 7, you another multiple of 7.

Back 31

35+21=56

Hint 31

(5x7)+(3x7) = (5+3)x7=8x7

Front 32

If you subtract two multiples of 7, you get another multiple of 7.

Back 32

35-21=14

Hint 32

(5x7)-(3x7)=(5-3)x7=2x7

Front 33

If you add or subtract multiples of N,

Back 33

the result is a multiple of N

Hint 33

OR
N is a divisor of x and of y, then N is a divisor of x+y

Front 34

Prime Number

Back 34

Any positive integer larger than 1 with exactly two factors. 1 & itself.

Front 35

The number or integer 2

Back 35

Is the ONLY EVEN prime number and also the first prime number.

Front 36

The first 10 prime numbers are?

Back 36

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

Front 37

If a problem states or assumes that a number is an integer...

Back 37

you MAY need to use prime factorization to solve the problem

Hint 37

1. Determining whether one number is divisible by another number
2. Determining the greatest common factor of two numbers
3. Reducing fractions
4. Finding the least common multiple of two or more numbers
5. Simplifying square roots
6. Determining the exponent on one side of an equation with integer constraints.

Front 38

Factor Foundation Rule

Back 38

If a is a factor of b, and bi is a factor of c, then a is a factor of c.

Hint 38

Other words:
An integer is divisible by all of its factors--and it is also divisible by all of the FACTORS of its factors.

Front 39

Greatest Common Factor (GCF)

Back 39

The largest divisor of two or more integers.

Front 40

Lease Common Multiple (LCM)

Back 40

The smallest multiple of two or more integers.

Front 41

Even Numbers

Back 41

Numbers divisible by 2

Front 42

Odd Numbers

Back 42

Not divisible by 2

Front 43

Add or subtract 2 odds or 2 evens and the result is

Back 43

EVEN

Front 44

Add or subtract an odd with an even, and the result is

Back 44

ODD

Front 45

When you multiply integers, if ANY of the integers is even, the result is

Back 45

EVEN

Front 46

When you multiply integers, if NONE of the integers is even, the result is

Back 46

ODD

Front 47

Odd +or- even

Back 47

ODD

Front 48

Odd +or- odd

Back 48

EVEN

Front 49

Odd x odd

Back 49

ODD

Front 50

Even x even

Back 50

EVEN (and divisible by 4)

Front 51

Even +or-Even

Back 51

EVEN

Front 52

Odd x even

Back 52

EVEN

Front 53

Even/Even

Back 53

Even, Odd & Non-Integer Possibilities

Front 54

Even/Odd

Back 54

Even & Non-Integer Possibilities

Front 55

Odd/Even

Back 55

Only Non-Integer Possibilities

Hint 55

CANNOT produce an integer because the odd number will never be divisible by the factor of 2 concealed within the even number.

Front 56

Odd/Odd

Back 56

Odd & Non-Integer Possibilities

Front 57

An ODD number divided by ANY other type of integer

Back 57

CANNOT produce and EVEN integer

Front 58

The sum of two prime numbers

Back 58

will be EVEN (sum of two odds)

Hint 58

Unless one of the Primes is the number 2.