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Commutative property of addition.

Commutative property of multiplication.

Associative property of addition.

Associative property of multiplication.

17+38=38+17

21x55=55x21

(4+5)+10=4+(5+10)

(5x3)x7=5x(3x7)

Commutative property of addition.

17+38=38+17

Commutative property of multiplication.

21x55=55x21

communitive property of addition

communitive property of multiplication

associative property of addition

associative property of multiplication

3 + 2 = 2 + 3

5 x 7 = 7 x 5

6 + (4 + 8) = (6 + 4) + 8

4 x (5 x 9) = (4 x 5) x 9

communitive property of addition

3 + 2 = 2 + 3

communitive property of multiplication

5 x 7 = 7 x 5

Closure

Closure

Associative

Commutative

Being in the same set after operating with numbers. Real numb…

If x and y and real #s, x+y is real, x•y is real

(x+y)+z=x+(y+z)

3+2=2+3

Closure

Being in the same set after operating with numbers. Real numb…

Closure

If x and y and real #s, x+y is real, x•y is real

Commutative property of addition

Commutative property of multiplication

Associative property of addition

Associative property of multiplication

7+3=3+7 (order doesn't matter, same sum)

3x7=7x3 (order doesn't matter, same product)

(2+3)+1=2(3+1) (doesn't matter what was added first)

(2x3)x1=2(3x1) (doesn't matter which are multiplied first)

Commutative property of addition

7+3=3+7 (order doesn't matter, same sum)

Commutative property of multiplication

3x7=7x3 (order doesn't matter, same product)

rules of divisibility by 2,3,4,5,6,8,9…

easiest way to check for divisibility…

if you add or subtract multiples on N,…

remember "fewer factors, more multiples"

an integer is divisible by... ... -2 if the integer is even ... -3…

just do long division

...

infinite multiples but only a few factors ... -factor aka a div…

rules of divisibility by 2,3,4,5,6,8,9…

an integer is divisible by... ... -2 if the integer is even ... -3…

easiest way to check for divisibility…

just do long division

Abundant

Amicable

Apocalyptic Power

Aspiring

The number n is abundant if the sum of all its positive divis…

The number n is amicable if it belongs to an amicable pair. T…

The number n is called an apocalyptic power if 2n contains th…

The number n is called an aspiring number if its aliquot sequ…

Abundant

The number n is abundant if the sum of all its positive divis…

Amicable

The number n is amicable if it belongs to an amicable pair. T…

Sum of consectuive numbers

Odd number of terms in set

5 consecutive integers will

Number series with 1 even

( first x last) ( last / 2)

Sum will be a multiple of the number of numbers

Always be divisible by 5 when you multiply them together

Multiple of 2

Sum of consectuive numbers

( first x last) ( last / 2)

Odd number of terms in set

Sum will be a multiple of the number of numbers

closure

commutative

associative

Identity

the sum or product of any two real numbers is a real number

order doesn't matter with addition or multiplication

grouping doesn't matter with addition or multiplication

there is a unique (one and only one) real number 0, and a uni…

closure

the sum or product of any two real numbers is a real number

commutative

order doesn't matter with addition or multiplication

Closure Property for Addition

Closure Property for Multipication

Commutative Property for Addition

Commutative Property for Multiplication

a + b is a rational number

ab is a rational number

a + b = b + a

ab = ba

Closure Property for Addition

a + b is a rational number

Closure Property for Multipication

ab is a rational number

Commutative Property of Addition

Associative Property of Addition

Identity property of addition

Additive Inverse Property

Says if you are adding two or more numbers, you can switch th…

Says that when you add 3 numbers, you can group the numbers i…

Says that no matter what number, it will stay the same if add…

When you add a number and its opposite, the result is zero... E…

Commutative Property of Addition

Says if you are adding two or more numbers, you can switch th…

Associative Property of Addition

Says that when you add 3 numbers, you can group the numbers i…

ODD number of negative signs

GMAT integer

Divisible by 6

Divisible by 9

Answer will be NEGATIVE

Non-fraction/non-decimal

Integer is divisible by BOTH 2 (even) and 3 (numbers added up…

SUM of integers digits are divisible by 9

ODD number of negative signs

Answer will be NEGATIVE

GMAT integer

Non-fraction/non-decimal

Reflexive Property

Symmetric Property

Transitive Property

Identity Property

equal to self... 5 = 5 ; 2+11 = 2+11

reflection... if x = y then y = x

a = b and b = c, then a = c also

doesn't change the value... addition: x + 0 = x... multiplication:…

Reflexive Property

equal to self... 5 = 5 ; 2+11 = 2+11

Symmetric Property

reflection... if x = y then y = x

Integer divisible by 3

Integer Divisible by 4

Integer Divisible by 6

Integer Divisible by 8

If the sum of the integer's digits is divisible by 3

If the integer is divisible by 2 twice, or if the last two di…

If the integer is divisible by both 2 and 3

If integer is divisible by 2 three times, or if the last thre…

Integer divisible by 3

If the sum of the integer's digits is divisible by 3

Integer Divisible by 4

If the integer is divisible by 2 twice, or if the last two di…

An integer is divisible by 2 if...

An integer is divisible by 3 if...

An integer is divisible by 4 if...

An integer is divisible by 5 if...

the integer is even

the sum of the integer's digits is divisible by 3

the integer is divisible by 2 twice OR if the last 2 digits a…

the integer ends in 0 or 5

An integer is divisible by 2 if...

the integer is even

An integer is divisible by 3 if...

the sum of the integer's digits is divisible by 3

Commutative Property

Reflexive Property

Substitution Property

Addition Property of Equality

a+b=b+a

for any real #a, a=a... 3=3 or a+b+2=a+b+2

if a=b, then b=a (hint: they switch across the equal side)... 6…

if a=b, then a+c=b+c

Commutative Property

a+b=b+a

Reflexive Property

for any real #a, a=a... 3=3 or a+b+2=a+b+2

commutative property of addition

commutative property of multiplication

associative property of addition

Associative property of multiplication

a+b = b+a Ex 2+3=3+2

ab=ba Ex 2 times 3 = 3 times 2

(a+b)+c = a+(b+c)

(ab)c=a(bc)

commutative property of addition

a+b = b+a Ex 2+3=3+2

commutative property of multiplication

ab=ba Ex 2 times 3 = 3 times 2

Integers

Rational numbers

Irrational numbers

PEMDAS

Whole numbers... Include negative whole numbers and zero

A number that can be expressed as a ratio of two integers

Real numbers that can be put on a number line but CANNOT be e…

Parenthesis ... Exponents... Multiplication ... Division... Addition... Subt…

Integers

Whole numbers... Include negative whole numbers and zero

Rational numbers

A number that can be expressed as a ratio of two integers

commutative

associative

additive identity

additive inverse

order does not matter when you add 1+2=2+1

grouping does not matter when you add (1+2)+3=1+(2+3)

(a numbers identity doesn't change when 0 is added) 2+0=2

the sum of the number and its additive inverse or opposite is…

commutative

order does not matter when you add 1+2=2+1

associative

grouping does not matter when you add (1+2)+3=1+(2+3)

Commutative

Associative

Additive Identity

Substitution

The order for adding or multiplying numbers does not matter;…

The terms in addition or multiplication problems can be assoc…

The value (identity) of a number, term, expression is not cha…

Replacing a variable in a term or expression with a number to…

Commutative

The order for adding or multiplying numbers does not matter;…

Associative

The terms in addition or multiplication problems can be assoc…

Comutative

Associative

Distributive

Transitive

Changing the order of numbers.... Addition ... Multiplication

Regrouping doesn't change the result... 2+(3+5) = (5+2)+3

The number in the outside is multiplied by the one in the ins…

If 3+5=a and 2+6=b than a=b

Comutative

Changing the order of numbers.... Addition ... Multiplication

Associative

Regrouping doesn't change the result... 2+(3+5) = (5+2)+3

Reflexive

Symmetric Property

Transitive Property

Substitution Property

Any quanity is equal to itself... 4+7=4+7

If one quantity equals a second then the second quantity equa…

If one quantity equals a second quantity and the second quant…

A quantity may be substituted for its equal in any expression…

Reflexive

Any quanity is equal to itself... 4+7=4+7

Symmetric Property

If one quantity equals a second then the second quantity equa…

When is an integer divisible by 3?

When is an integer divisible by 4?

When is an integer divisible by 6?

When is an integer divisible by 8?

Sum if the digits is divisible by 3

Divisivle by 2 twice, or the last 2 digits are divisible by 4

Divisible by 2 and 3

Divisible by 2 three times, or last 3 digits divisible by 8

When is an integer divisible by 3?

Sum if the digits is divisible by 3

When is an integer divisible by 4?

Divisivle by 2 twice, or the last 2 digits are divisible by 4

Distributive Property

Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Addition

Changing the Order... a+b = b+a

Changing the Order... ab = ba

Moving Parentheses - keep order the same... (a+b)+c = a+(b+c)

Distributive Property

Commutative Property of Addition

Changing the Order... a+b = b+a

Prime numbers less than 20

Prime numbers between 20-50

Divisible by 2

Divisible by 3

2,3,5,7,9,11,13,27,19

23, 29, 31, 37, 41, 43, 47

The integer is even

The sum of the integer's digits is divisible by 3

Prime numbers less than 20

2,3,5,7,9,11,13,27,19

Prime numbers between 20-50

23, 29, 31, 37, 41, 43, 47

Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Additon

Associative Property of Multiplication

a + b + c = c + b + a (move; change order) 2 + 3 + 4 = 4 + 2 +3

abc= cba (changing order) (9)(8)(7) = (8)(7)(9)

x + (y+z) = y + (x + z) (changing grouping) 5 + (10 + 15) = 1…

m ** (n**p) = p** (m**n) (changing grouping) 2 (5 ** 7) = 7 (2 ** 5)

Commutative Property of Addition

a + b + c = c + b + a (move; change order) 2 + 3 + 4 = 4 + 2 +3

Commutative Property of Multiplication

abc= cba (changing order) (9)(8)(7) = (8)(7)(9)

Commutative Property of Addition

Associative Property of Addition

Identity Property of Addition

Inverse Property of Addition

Numbers can be added in any order; the sum will always be the…

The grouping of numbers being added does not matter the sum w…

If zero is added to any number the sum will be that number

The sum of a number and its opposite will be zero

Commutative Property of Addition

Numbers can be added in any order; the sum will always be the…

Associative Property of Addition

The grouping of numbers being added does not matter the sum w…

Associative Property of Addition

Associative Property of Multiplication

Commutative Property of Addition

Commutative Property of Multiplication

(a + b) + c = a + (b + c)

(a ∙ b) ∙ c = a ∙ (b ∙ c)

a + b = b + a

a ∙ b = b ∙ a

Associative Property of Addition

(a + b) + c = a + (b + c)

Associative Property of Multiplication

(a ∙ b) ∙ c = a ∙ (b ∙ c)

commutative property

associative property

distributive property

additive identity

the order of the numbers does not affect the sum or product

the grouping of the numbers does not affect the sum or product

multiplication can be distributed over addition or subtraction

when zero is added or subtracted from a number, it remains th…

commutative property

the order of the numbers does not affect the sum or product

associative property

the grouping of the numbers does not affect the sum or product

Associative Property (words)

Associative Property (+)

Associative Property (x)

Commutative Property (words)

In addition or multiplication, numbers can be grouped in any…

(a + b) + c = a + (b + c)

(a×b)×c = a×(b×c)

In addition and multiplication, numbers can be computed in an…

Associative Property (words)

In addition or multiplication, numbers can be grouped in any…

Associative Property (+)

(a + b) + c = a + (b + c)

Commutative Property of Addition

Commutative Property of Multiplication

Associative Property of Addition

Associative Property of Multiplication

6 + 5 = 5 + 6

(-9)(3) = 3(-9)

(3 + 2) + 5 = 3 + (2 + 5)

-4(2 x 8) = (-4 x 2) x 8

Commutative Property of Addition

6 + 5 = 5 + 6

Commutative Property of Multiplication

(-9)(3) = 3(-9)