Number Properties

Front 1

Absolute Value

Back 1

Numbers distance on the # line.

(Ex: |-3| = 3 v |3| = 3)

Front 2

How do u set-up D/S?

Back 2

AD/BCE or BD/ACE

Front 3

Sum of Integers:

20 to 100 inclusive?

(X to Y Inclusive)

Back 3

(1) Avg: (20+100)/2= 60

(2) Number of Terms: (100-20)+1=81

(3) Sum of Terms: (Average) * (# of Terms) = 4,860

Front 4

Average of Integers:

20 to 100 inclusive?

Back 4

(100 + 20)/2 = 60

Front 5

How many integers btw. 10 to 30 inclusive?

(X to Y Inclusive?)

Back 5

30-10+1=21

(y-x+1=solution)

Front 6

Will avg. of even number of consecutive terms be an integer? Why?

Back 6

No, because there is no middle number in an even set of consecutive numbers.

Front 7

Multiple

Back 7

Product of a specified number and an integer.

Ex: Multiples of 3-
3(1)=3, 3(4)=12, 3(90)=90

Front 8

If 72 is divisible by 12, is 72 divisible by all the factors of 12? Why?

Back 8

Y, Factor Foundation Rule

Factors are the foundation upon which all numbers are built.

Front 9

If a problem involves too much competition is there probably an easier way to solve?

Back 9

Yes

Front 10

If there is one even integer in a consecutive series. Product of the series is divisible by?

Back 10

2

Front 11

Prime numbers between ( 0 to 50)

How many numbers?

Back 11

2,3,5,7,11,13,17,19,23,29,31,37,
41,43,47

How many? 15 numbers

Front 12

If 2 #'s have the same divisor then____?

Back 12

The sum and difference of the numbers are also divisible by the #.

Front 13

Prime Factors of 10

Back 13

2, 5

Front 14

If "x" is divisible by "5". Should u assume "5" is "X"s only divisor? Why?

Back 14

No, there can be other #'s in the prime-box.

Front 15

What is a non-prime factor of all integers?

Back 15

1

Front 16

Smallest Prime

Back 16

2

Front 17

When testing in D.S. what are you trying to prove?

Back 17

The statement is insufficient

Front 18

What should you test in D.S.?
Unless?

Back 18

Fractions, negatives and Zero

Unless- told variables only represent: (1) integers, (2) positive numbers or (3) non-zero numbers

Front 19

How many solutions do absolute value questions have?

Back 19

2

Front 20

What is both a factor & multiple of itself? Why?

Back 20

Ans: Integer

Why?
Take Random Integer: "8"

Factor: Number divides evenly into an integer. (8/8=1) Therefore, "8" is a factor of "8"

Multiple: Formed by multiplying that integer by any whole number. (8*1)=8. Therefore, "8" is a multiple of "8".

Front 21

The easier the method to solve a problem________?

Back 21

1. Faster to solve
2. Less chance of making mistake

Front 22

Special sum rule?

Back 22

Sum of odd # of consecutive integers will be a multiple of that # of odd integers.

Example: Sum of "3" odd integers is a multiple of "3".

Front 23

Properties of "-1"

Back 23

Multiplying or dividing a nonzero # by "-1" changes the sign of the #.

Front 24

√x=

Back 24

X½

Front 25

even/odd=

Back 25

even (*If an integer)

Front 26

√a + √b ≠

Back 26

√a+b

Front 27

How do you reveal the mathematical content and info disguised in data sufficiency problems?

Back 27

Rephrase

Front 28

LCM?

LCM of 30 and 24?

Back 28

Smallest number that is a multiple of two integers

Prime Factors: 30
[2, 3, 5]

Prime Factors: 24
[2, 2, 2, 3]

LCM= product of all prime factors of both number, using higher power of repeated factors.

LCM= 2*2*2*3*5=120

Front 29

GCF?

GCF of 30 and 24?

Back 29

Largest number by which two integers can be divided

Prime Factors: 30
[2, 3, 5]

Prime Factors: 24
[2, 2, 2, 3]

GCF= product of common prime factors, using lower power of the repeated factor.

GCF= 2*6=6

Front 30

√16?

Back 30

-4 v 4

Front 31

Relationship between all even exponents & squares

Back 31

All even exponents are squares

Front 32

Integer

Back 32

Number w/o fractional or decimal parts

Front 33

Can negative numbers be integers?

Back 33

Yes

Front 34

Can whole numbers be integers?

Back 34

Yes

Front 35

Is zero an integer?

Back 35

Yes

Front 36

Prime Number?

Back 36

1. Integer>1
2. Two Factors: Itself & "1"

Front 37

Why is "1" not prime?

Back 37

Only divisible by itself

Front 38

An integer is "divisible" by a number if__________?

Back 38

If the integer can be evenly divided by that number.

Ex: 15 is divisible by 3

Front 39

Integer divisible by "2"

Back 39

Last # is divisible by "2"

Front 40

Integer divisible by "3"

Back 40

If digits add up to a multiple of "3"

Front 41

Integer divisible by "8"

Back 41

Half the number "3" times

Front 42

Positive/Positive

Back 42

Positive

Front 43

If u see a question w/ word "remainder" in it- generally testing?

Back 43

Divisibility

Front 44

Fraction

Back 44

Part/Whole

Example: 2/4

Front 45

Connection between divisibility & multiples

Back 45

If a said number is evenly divisible by another- that number is also a multiple of that number

Ex: 54/4=13

1. 52 is divisible by 4
2. 52 is also a multiple of 4

b/c: 4(13)= 52

Front 46

X% of Y =

10% of 20=

Back 46

Y% of X

20% of 10

Front 47

Raising a "negative" number to an "even" power produces _____?


(-2)²=

Back 47

Positive Result

(-2)²=4

Front 48

Raising a "negative" number to an "odd" power produces _____?


(-2)³=

Back 48

Negative Result

(-2)³= -8

Front 49

Can you take the even root of a negative number?

√X, when X= -3

Back 49

No

Front 50

Can you take the odd root of a negative number?


3√X, when X= -3

Back 50

Yes

Front 51

Strategy for "%" problems w/ fractions in answer choice?

Back 51

If there is a denominator that is common in most answer choices, use it as the base #.

Front 52

4√X^52 =?

Back 52

Divide exponent by "4"

52/4= 13

4√X^52 =X^13

b/c:

X^52=(X^13)(X^13)(X^13)(X^13)

Front 53

What is the square root of any even exponent?

Ex: √X^52 or √X^76

Back 53

Base Raised 1/2 Exponent

√X^52= X^26

[b/c (X^26)(X^26)=X^52]


√X^76= X^38

[b/c (X^38)(X^38)=X^76]

Front 54

What is x²-2x or x²+2x?

Even or odd?

Back 54

Product of consecutive integers.

Odd or Even- depending on value of "X"
_____________
1. OxO=O
2. ExE= E

_____________

1. x²-2x= x(x-2)

2. x²+2x= x(x+2)

Front 55

Square Root √289?

[MG Strategy]

Back 55

1. Number btw 15² & 20²

2. What digit squared btw 5² to 9², yield units digit 9?

3. 7²=49

4. Therefore, answer is "17"

Front 56

Square Root √576?

[MG Strategy]

Back 56

1. Number btw 20² & 25²

2. What digit squared btw 0² to 5², yield units digit 6?

3. 4²=16

4. Therefore, answer is "24"

Front 57

If avg. of integers (1 to x) inclusive is 11. How many numbers in the set?

Back 57

1. (1+x)/2=11

2. x=22-1

3. x=21

Front 58

How do you express "2" unknown variables?

Back 58

Express 2nd term using variable from 1st term.

Ex:
1st Term: y
2nd Term: y-7

Front 59

1+2+3+4=

Back 59

10

Front 60

6+7+8+9+10=

Back 60

40

Front 61

Sum of Terms

Back 61

number of terms x average

Front 62

You can combine exponents linked by______?

Back 62

Multiplication or Division

Front 63

Why are even exponents dangerous?

Back 63

Hides the sign of the base

b²=4 [b= -2v2]

Front 64

Profit or discount expressed as a %?

Back 64

[(Profit or Discount)/Original Price] *100%

Front 65

When will odds be even? (3)

Back 65

1. O-O=E
2. O+O=E
3. OxE= E

Front 66

-1/2º=

Back 66

1

Front 67

What is x²-x or x²+x?

odd or even?

Back 67

Product of consecutive integers

1. x²-x= x(x-1)

2. x²+x= x(x+1)
________

Always "even"

Front 68

Quadratic Form: #3

1. x²-2xy+y²

Ex:
x²-6x+9

Back 68

1. (x-y)²= (x-y)(x-y)
_______________

Ex:
(x-3)(x-3)

Front 69

"N" is divisible by 3, 7 and 11

What other numbers must be divisors of "N"?

Back 69

Since 3,7 and 11 are all prime factors → "N" must also be divisible by all possible prime products
____________

Ans: 21, 33, 77, 231

Front 70

Simplify:

(x^a)/(x^b)=

Back 70

x^a-b

Front 71

Simplify:

(√x)/(√y)

Back 71

√x/y

Front 72

Simplify:

√a²

Back 72

√a * √a = a

Front 73

Simplify:

(X^a)*(X^b)

Back 73

X^a+b

Front 74

Simplify:

X^1=

3^1=

Back 74

X^1=X

3^1=3
_________
Raise any exponent to "1", keep its base

Front 75

Simplify:

6^5-6^3

Back 75

Cannot simplify

Front 76

Does:

√16 + √9= √16+9

Back 76

No, cannot add exponents

Front 77

Simplify:

7^4-7^6

Back 77

Cannot simplify

Front 78

Does addition come before subtraction?

Back 78

Not necessarily, work problem from "left" to "right"

Front 79

Does mult. come before division?

Back 79

Not necessarily, work problem from "left" to "right"

Front 80

Can you combine or split? Why?

√64 * √25 "or"

√25/4

Back 80

Yes, mult. & division

Front 81

Sum of "2" primes?

Back 81

"E", expect one of them is "2".

Front 82

Odd-Odd=

Back 82

Even

Front 83

Even-Odd=

Back 83

Odd

Front 84

Even * Even=

Back 84

Even

Front 85

Odd * Even=

Back 85

Even

Front 86

Odd * Odd=

Back 86

Odd

Front 87

Odd + Odd=

Back 87

Odd

Front 88

Def. Quadratic Equation?

Back 88

ax²+bx+c=0

Front 89

Even - Even=

Back 89

Even

Front 90

Simplify:

9³/3³=

Back 90

3³

Front 91

Simplify:

3² * 3³=

Back 91

3^5

Front 92

Simplify:

3³ * 5³=

Back 92

15³

Front 93

Raising a fraction between (0-1) to a power produces a ____ result.

(1/2)²=

Back 93

Smaller

(1/2)²: (1/2)*(1/2)= 1/4

Front 94

How do you know if "4" is a factor of "12"?

Back 94

Prime Factorization:

Can you create "4" from the prime factors of "12"?

Prime Factors of 12:
2,2,3

Therefore, answer is "yes"

Front 95

How do you determine all the factors of a #?

Back 95

Prime Factorization:

"Prime Box"
*Build products of all prime factors

Front 96

Simplify:

(3^5)/(3^3)

Back 96

3²

Front 97

Even Number

Back 97

Divisible by "2"

Front 98

a√c + b√c=

Back 98

a+b√c

Front 99

Prime Factors: 8

Back 99

2,2,2

Front 100

Prime Factors: 9

Back 100

3,3

Front 101

Even * Even=

Back 101

Even

Front 102

Prime Factors: 12

Back 102

2,2,3

Front 103

Odd Integers

Digits: ?

Back 103

Integers not divisible by "2"

Digits: 1,3,5,7,9

Front 104

Simplify:

√1919 / √.1919

Back 104

1. (√10,000 * √.1919)/ √.1919=

[*√.1919 cancels out]

2. √10,000= 100

Front 105

Quadratic Form: #1

x²-y²=
_________

Ex:
x²-9=

Back 105

1. (x+y)(x-y)

Ex:
(x+3)(x-3)

Front 106

Quadratic Form: #2

x²+2xy+y²=
___________

Ex:
x²+6x+9=

Back 106

1. (x+y)²=(x+y)(x+y)

Ex:
(x+3)²= (x+3)(x+3)

Front 107

Simplify:

(Xª)^b=

Back 107

x^a*b

Front 108

Divisible by 12

Back 108

Divisible by 3 & 4

Front 109

If a # is divisible by "2" or more #'s, is it divisible by the product of those "2" #'s? LCM of those "2" #'s?

Back 109

Ex: 12

Divisible by- 4 & 6

LCM=12
Product=24

Therefore: Always divisible by LCM, not necessarily product (unless the #'s are prime and results in a prime product.)

Front 110

Simplify:

3² * 3³=

Back 110

3^5

Front 111

Simplify:

3^3 * 5^3=

Back 111

15^3

Front 112

If "2" even integers in a consecutive series. Product of series divisible by?

Back 112

4

Front 113

Real Numbers

Back 113

All integers & everything in-between

Front 114

Odd * Odd=

Back 114

Odd

Front 115

Will the average of odd # of consecutive terms be an integer? Why?

Back 115

Yes, b/c its the middle #

Front 116

If # divisible by "4" and "6", is it divisible by "24"?

Back 116

No divisible by LCM ≠ product, unless product derived by prime factors.

Front 117

Simplify:

x³-x=

What is it?

Back 117

Product of 3 Consecutive Numbers:

x(x²-1)= x(x+1)(x-1)

= (x-1)(x)(x+1)

Front 118

Odd + Even=

Back 118

Odd

Front 119

Simplify:

√x * √y=

Back 119

√xy

Front 120

"Prime box" useful for solving what kind of problems? (3)

Back 120

1. GCF
2. LCM
3. Divisibility

Front 121

Digits

Back 121

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

Front 122

Simplify:

x-y=

Back 122

(√x+√y)(√x-√y)

Front 123

R all prime #'s odd?

Back 123

Y, expect "2"

Front 124

2 ways to deal w/ imperfect square

Ex: √52

Back 124

1. Estimate

√49= 7 & √64= 8
Therefore: √52 btw 7&8 ≈7.2

2. Factor out perfect square

√4*13
=2√13

Front 125

Multiplication is repeated______

Back 125

Addition

Front 126

How can an integer have exactly three positive divisors or factors?

Back 126

Must be the perfect square of a prime number

Front 127

Raising one exponent to another is repeated_________

Back 127

multiplication

Front 128

Simplify:

1/x^y=

Back 128

x^-y

Front 129

Simplify:

x*x=

Back 129

x²

Front 130

5+6+7+8+9=

Back 130

35

Front 131

Average

Back 131

Sum of Terms/# of Terms

Front 132

Distinguish btw "value" & "yes/no" data sufficiency

Back 132

Value: "1" answer

Yes/No: answering "yes" or "no"
is sufficient
[*maybe= non-sufficient]

Front 133

Prime Factors: 4

Back 133

2,2

Front 134

Prime Factors: 6

Back 134

2, 3

Front 135

Divisible by "4"

Back 135

1. Half the number "twice"

2. Last "2" digits multiple of "4"

Front 136

Divisible by "5"

Back 136

Last digit "0" or "5"

Front 137

Divisible by "6"

Back 137

Divisible by "2" and "3"

Front 138

"2" defining characteristics of a quadratic

Back 138

1) 1st term raised to 2nd power

2) 2nd term raised to 1st power

Front 139

What does this expression mean?

3√64

Back 139

What number when multiplied by itself "3" times equates to "64"

3√64= 4

b/c: 4³=64

Front 140

How many "multiples" of 7 from 8 to 49 inclusive?

Back 140

1. List all the multiples

14,21,28,35,42,49 = 6 multiples

"or"

2. "7" multiples of "7" = 49 (7²=49)

Therefore, since 7^1 not included = 6 multiples

Front 141

Simplify:

(3²)^4

Back 141

3^8

Front 142

How can rounding help on the GMAT?

Back 142

1. Rounding can save time

2. Make sure answer choices are not extremely close

Front 143

Simplify:

3/(√6+√5)

Back 143

1. Multiply top and bottom to rid of radicals

3(√6-√5)/√6+√5(√6-√5)

2. (3√6-3√5)/6-5=

=3√6-3√5

Front 144

Simplify:

(x^22 - y^18)/ (x^11 + y^9)

Back 144

1.
(x^11 + y^9) (x^11 - y^9)/ (x^11 + y^9)

2. = x^11 - y^9

Front 145

What is the GMAT's greatest trick w/ exponents?

Back 145

Hiding the sign of an even exponent.

x²=4

x= 2 v -2

Front 146

Simplify:

(x^2)(X^-2)

Back 146

1) Add the exponents

2 + -2= 0

2) (x^2)(X^-2)= xº =1

Front 147

Pemdas

Back 147

Parentheses

Exponents

Mult

Division

Add

Sub

Front 148

Divisible by "9"

Back 148

Digits add to mult. of "9"

Front 149

What is special product rule?

Back 149

Product of any number of consecutive #'s will be divisible by that # of consecutive integers

Ex: Product of 10 consecutive integers divisible by 10

Front 150

Even + Even=

Back 150

Even

Front 151

How does GMAT attempt you trick you w/ roots?

Back 151

Combine or split roots dealing w/ addition or subtraction

Front 152

Odd - Even=

Back 152

Odd

Front 153

Even + Even=

Back 153

Even

Front 154

Simplify:

x^22 - y^18=

Back 154

(x^11 + X^9)(X^11 - X^9)

Front 155

Simplify:

x / 0=

Back 155

undefined

Front 156

Is:

x/yz = 1/y(x/z)

"or"

x/yz = 1/z(x/y)

Back 156

Both

Front 157

On the GMAT are you supposed to solve problems like:

(n+2)(n+3)(n+4)=990

What are you supposed to do?

Back 157

No, supposed to estimate

Hint: 990 close to 1,000
Therefore, one of the numbers is probably 10. Test #'s realize consecutive #'s are: 9, 10, 11

Front 158

Simplify:

x-y

Back 158

(√x + √y) (√x - √y)

Front 159

Quick Divisibility:

Is 443 divisible by "7"?

Back 159

1. 6 * 7= 42

2. 42 * 10= 420

3. 443-420= 23

4. 23/7≠ integer

Therefore: ans "no"

Front 160

What is the value of:

(x^2+x^-2)² - (x^4+x^-4)

1. 0
2. 1
3. 2
4. 3
5. 1,2,3

Back 160

1. (x²+x-²)=
(x²+x-²)(x²+x-²)=
(x²)(x²)+ 2(x²)(x-²)+ (x-²)(x-²)=
x^4+ 2(1)+ x^-4

2. Therefore

(x^2 + x^-2)² -(x^4+x^-4)=

x^4+x^-4+2 -(x^4+x^-4)= 2