Number Properties Flashcards

1
Q

What are integers?

A

Integers are “whole” numbers such as 0,1,2 and 3 that have no fractional part.

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2
Q

Integers can be positive, negative, or zero?

A

All are correct, integers can be positive (1,2,3…), negative (-1,-2,-3…), or the number 0.

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3
Q

The GMAT uses the term integer to mean a ______ or a ______.

A

The GMAT uses the integer to mean a NON FRACTION or a NON DECIMAL.

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4
Q

The sum of 2 integers is always an integer. T or F?

A

TRUE. The sum of 2 integers is ALWAYS an integer.

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5
Q

The difference of two integers is always and integer - True or False?

A

TRUE. The difference of 2 integers is ALWAYS an integer.

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6
Q

The product of 2 integers is never an integer. T or F?

A

FALSE. The product of 2 integers is ALWAYS an integer.

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7
Q

The result of dividing two integers is NOT an integer.

A

FALSE. It’s SOMETIMES an Integer.

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8
Q

Divisible by 2?

A
  • 2, integer is EVEN
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9
Q

Divisible by 3?

A
  • 3, SUM of integers digits is divisible by 3
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10
Q

Divisible by 4?

A
  • 4, divisible by 2 twice OR if last 2 digits are divisible by 4
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11
Q

Divisible by 5?

A
  • 5, integer ends in 0 or 5
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12
Q

Divisible by 6?

A
  • 6, divisible by both 2 AND 3
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13
Q

Divisible by 8?

A
  • 8, divisible by 2 three times OR last 3 digits divisible by 8
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14
Q

Divisible by 9?

A
  • 9, sum of digits divisible by 9
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15
Q

Divisible by 10?

A
  • 10, number ends in 0
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16
Q

Factors and Multiples are essentially WHAT terms?

A

Opposite

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17
Q

What are factor pairs?

A
  • pairs of factors that yield an integer when multiplied together
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18
Q

How to find a factor pair of X?

A
  1. Make a table with 2 columns labeled “small” and “large”
  2. Start with 1 in the small column and X in the large
  3. Test the next possible factor of X
  4. Repeat until the numbers in the small and large columns converge
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19
Q

What is a Factor?

A

Positive integer that divides evenly into an integer.

  • every integer is a factor of itself
  • 1 is a factor of every integer
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20
Q

What is a Multiple?

A

Integer formed by multiplying that integer by any integer.

  • negative multiples are possible
  • zero is a multiple of every number
  • every integer is a multiple of itself
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21
Q

What is the mnemonic to not confuse FACTORS and MULTIPLES?

A

“Fewer Factors, More Multiples”

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22
Q

Do all of the following say the same thing:

  • X is divisible by Y = Y is a divisor of X
  • X is a multiple of Y = Y divides X
  • X/Y is an integer = X/Y yields a remainder of 0
  • X=3(n), n being an integer = Y “goes into” X evenly
A

Yes!

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23
Q

If you add or subtract Multiples of N, the result is?

A

a multiple of N.

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24
Q
  • N is the divisor of x and y, then N is a divisor of WHAT?
A

divisor of x + y

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25
Q

What is Prime?

A

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

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26
Q

is 1 Prime?

A

1 is NOT considered a prime

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27
Q

What is the first prime? And why is it special?

A
  • the first prime is 2, only even prime
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28
Q

first ten primes are…

A

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

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29
Q

How to find primes?

A

To find primes, create a prime factor tree.

- factors of N can be found by building all possible products of the prime factors

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30
Q

When to use Prime Factorization?

A
  1. Determine if x is divisible by y
  2. Determine the greatest common factor of two numbers
  3. Reducing fractions
  4. Finding the least common multiple of a set of numbers
  5. simplifying square roots
  6. Determine the exponent of one side of an equation with integer constraints
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31
Q

What is the Factor Foundation Rule?

A

If A is a factor of B, and B is a factor of C, then A is a factor of C.

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32
Q

What is a Prime Box?

A

Holds all prime factors of N.

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33
Q

What can you tell by multiplying factors in a prime box

A

can tell if X is a factor of Y by multiplying factors in the prime box

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34
Q

What is the Greatest Common Factor (GCF)

A

GCF: the largest divisor of 2+ integers

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35
Q

Least Common Multiple (LCM)

A

LCM: the smallest multiple of 2+ integers

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36
Q

If no primes in common, what is the GCF and LCM?

A
  • if no primes are in common, the GCF is 1 and the LCM is the product of the two numbers
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37
Q

On simpler remainder problems, what number is it best to pick?

A

On simple problems, pick numbers.

  • add the desired remainder to a multiple of the divisor
  • ex: need a number that leaves a remainder of 4 after dividing by 7; (7*2) + 4 = 18
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38
Q

Even +/- Even

A

Even

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39
Q

Even +/- Odd

A

Odd

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40
Q

Odd +/- Even

A

Odd

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41
Q

Odd +/- Odd

A

Even

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42
Q

Even * Even

A

Even

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43
Q

Even * Odd

A

Even

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44
Q

Odd * Even

A

Even

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45
Q

Odd * Odd

A

Odd

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46
Q

Divisibility of Odds & Evens

A

NO GUARANTEES!

  • can result in odds, evens or non-integers
  • odd divided by any number will never be even
  • odd divided by even will never equal an integer
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47
Q

What should 3 things should you remember when you see the sum of 2 primes

A

ALL primes are odd except for 2

  • sum of any two primes will always be even unless one of the primes is 2
  • if you see a sum of two primes that is odd, one number must be 2
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48
Q

What does absolute value tell you

A

Tells how far a number is from 0 on the number line and is always positive.

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49
Q

if two numbers are opposite each other, what do they have

A

if two numbers are opposite each other, they have the same absolute value

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50
Q

Multiplying & Dividing Signed Numbers

A

If Signs are the Same, the answer is positive but if Not, the answer is Negative.

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51
Q

Evenly Spaced Sets are…

A

Sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next
- ex: 4,7,10,13,16

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52
Q

Consecutive Multiples are…

A

Special cases of evenly spaced sets: all values in the set are multiples of the increment

  • ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4
  • these sets must be composed of integers
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53
Q

Consecutive Integers are…

A

Special cases of consecutive multiples: all the values in the set increase by 1, all integers are multiples of one
- ex: 12,13,14,15,16

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54
Q

What does absolute value tell you

A

Tells how far a number is from 0 on the number line and is always positive.

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55
Q

if two numbers are opposite each other, what do they have

A

if two numbers are opposite each other, they have the same absolute value

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56
Q

Multiplying & Dividing Signed Numbers

A

If Signs are the Same, the answer is positive but if Not, the answer is Negative.

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57
Q

Evenly Spaced Sets are…

A

Sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next
- ex: 4,7,10,13,16

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58
Q

Consecutive Multiples are…

A

Special cases of evenly spaced sets: all values in the set are multiples of the increment

  • ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4
  • these sets must be composed of integers
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59
Q

Consecutive Integers are…

A

Special cases of consecutive multiples: all the values in the set increase by 1, all integers are multiples of one
- ex: 12,13,14,15,16

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60
Q

What does absolute value tell you

A

Tells how far a number is from 0 on the number line and is always positive.

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61
Q

if two numbers are opposite each other, what do they have

A

if two numbers are opposite each other, they have the same absolute value

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62
Q

Multiplying & Dividing Signed Numbers

A

If Signs are the Same, the answer is positive but if Not, the answer is Negative.

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63
Q

Evenly Spaced Sets are…

A

Sequences of numbers whose values go up or down by the same amount (increment) from one item in the sequence to the next
- ex: 4,7,10,13,16

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64
Q

Consecutive Multiples are…

A

Special cases of evenly spaced sets: all values in the set are multiples of the increment

  • ex: 12,16,20,24 - increase by 4s, ea. element a multiple of 4
  • these sets must be composed of integers
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65
Q

Consecutive Integers are…

A

Special cases of consecutive multiples: all the values in the set increase by 1, all integers are multiples of one
- ex: 12,13,14,15,16

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66
Q

All sets of consecutive integers are sets of

A

consecutive multiples

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67
Q

All sets of consecutive multiples are

A

evenly spaced sets

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68
Q

All evenly spaced sets are fully defined if these three parameters are known:

A
  1. The smallest (first) or largest (last) number in the set
  2. The increment (always 1 for consecutive integers
  3. The number of items in the set
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69
Q

Arithmetic mean

A

= median.

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70
Q

if there are an even number of elements, the median is

A

the average of the two middle elements

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71
Q

Mean & median =

A

average of the first and last element.

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72
Q

Sum of elements =

A

mean * number of items in the set.

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73
Q

Consecutive integers formula:

A

(Last - First) + 1 “add one before you’re done”

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74
Q

Consecutive multiples formula:

A

(Last - First)/Increment + 1

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75
Q

Sum of a set of consecutive integers equals WHAT

A

The number of itmes in the set times the middle number (aka median of the set)

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76
Q

The product of K consecutive integers is always divisible by

A

The product of K consecutive integers is always divisible by K factorial (K!)
- ex: 3! = 3x2x1 = 6, always divisible by 3&2

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77
Q

For any set of consecutive integers with an ODD number of items, the sum of all integers is

A

ALWAYS a multiple of the number of items.

- ex: 1+2+3+4+5=15, multiple of 5

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78
Q

for any set of consecutive integers with an EVEN number of items, the sum of all items is

A

NEVER a multiple of the number of items.

- ex: 1+2+3+4=10, not a multiple of 4

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79
Q

What can you use to keep track of factors of consecutive integers.

A

Use prime boxes

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80
Q

What to do with a negative base with exponents

A

when negative, simply multiply as required

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81
Q

Why beware when there is an EVEN exponent

A

an EVEN exponent.

- hides the sign of the base - any base raised to an even power is a positive answer

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82
Q
  • odd exponents always keep the sign of the
A

base

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83
Q

base of 0=

A

=0

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84
Q

base 1=

A

=1

85
Q

base -1=

A

=1 when the exponent is even or -1, when the exponent is odd.

86
Q

Fractional: if positive proper fraction, as the exponent increases, the value of the expression

A

decreases

87
Q

increasing powers cause fractions to

A

decrease

88
Q

What do you do when (2x5)^3

A
  • when a product, multiply base first then raise to power OR distribute the exponent first then multiply
89
Q

When multiplying two numbers with the same base, combine exponents by

A

adding

90
Q

When dividing two numbers with the same base, combine exponents by

A

subtracting them

91
Q

When raising a power to a power, combine exponents

A

by multiplying

92
Q

when there is a negative exponent, think

A

reciprocal

93
Q

When see exponent of 1

A

ny number that doesn’t have an exponent implicitly has an exponent of 1

94
Q

any non-zero base raised to the power of 0 is equal to …

A

1

95
Q

In fractional exponents…what does the numerator and denominator tell us

A

the numerator tells us the POWER to raise the base to, and the denominator tells us which ROOT to take

96
Q

You can Simplify Exponential Expressions, that are linked by

A

Only if expressions are linked by multiplication or division.

97
Q

CANNOT simplify expressions linked by

A

addition or subtraction.

98
Q

if linked by multiplication or division if they, can only simplify if they have what 2 things

A

if they have a base or exponent in common.

99
Q

Factor whenever bases are WHAT

A

Factor whenever bases are the same

100
Q

Factor when the exponent is the same and the terms have something in

A

COMMON

101
Q

When GMAT does a square root or an even root, they are only looking for

A

the positive root.

102
Q

For odd roots, the answer will be the same sign as

A

the base.

103
Q

Even roots only have a WHAT value

A

positive

104
Q

When can you combine of simplify roots AND when can you not

A

Combining/separating in multiplication and division, NEVER in addition or subtraction

105
Q

How to simplify roots?

A
  • multiplying or dividing: split into factors or simplify into a single root of the product.
106
Q

How to Estimate Roots of Imperfect Squares

A
  1. If a simple square root (no coefficient), figure out two closest perfect squares on either side and estimate between those roots.
  2. If a coefficient, estimate as step one and multiply by coefficient OR combine coefficient with the root and then do step 1
107
Q

A

1

108
Q

1.4²

A

≈2

109
Q

1.7²

A

≈3

110
Q

2.25²

A

≈5

111
Q

A

4

112
Q

A

9

113
Q

A

16

114
Q

A

25

115
Q

A

36

116
Q

A

49

117
Q

A

64

118
Q

A

81

119
Q

10²

A

100

120
Q

11²

A

121

121
Q

12²

A

144

122
Q

13²

A

169

123
Q

14²

A

196

124
Q

15²

A

225

125
Q

16²

A

256

126
Q

20²

A

400

127
Q

25²

A

625

128
Q

30²

A

900

129
Q

√1

A

1

130
Q

√2

A

≈1.4

131
Q

√3

A

≈1.7

132
Q

√5

A

≈2.25

133
Q

√4

A

2

134
Q

√9

A

3

135
Q

√16

A

4

136
Q

√25

A

5

137
Q

√36

A

6

138
Q

√49

A

7

139
Q

√64

A

8

140
Q

√81

A

9

141
Q

√100

A

10

142
Q

√121

A

11

143
Q

√144

A

12

144
Q

√169

A

13

145
Q

√196

A

14

146
Q

√225

A

15

147
Q

√256

A

16

148
Q

√400

A

20

149
Q

√625

A

25

150
Q

√900

A

30

151
Q

A

1

152
Q

A

8

153
Q

A

27

154
Q

A

64

155
Q

A

125

156
Q

³√1

A

1

157
Q

³√8

A

2

158
Q

³√27

A

3

159
Q

³√64

A

4

160
Q

³√125

A

5

161
Q

PEMDAS

A
Parentheses, 
Exponents,
Multiplication, 
Division, 
Addition, 
Subtraction
162
Q

One of the most common errors involving orders of operations occurs when an expression with multiple terms is ____

A

Subtracted

163
Q

What should you pretend there are around the numerator and denominator of a fraction

A

Parentheses

164
Q

A

27

165
Q

A

64

166
Q

A

125

167
Q

³√1

A

1

168
Q

³√8

A

2

169
Q

³√27

A

3

170
Q

³√64

A

4

171
Q

³√125

A

5

172
Q

PEMDAS

A
Parentheses, 
Exponents,
Multiplication, 
Division, 
Addition, 
Subtraction
173
Q

√1

A

1

174
Q

√2

A

≈1.4

175
Q

√3

A

≈1.7

176
Q

√5

A

≈2.25

177
Q

√4

A

2

178
Q

√9

A

3

179
Q

√16

A

4

180
Q

√25

A

5

181
Q

√36

A

6

182
Q

√49

A

7

183
Q

√64

A

8

184
Q

√81

A

9

185
Q

√100

A

10

186
Q

√121

A

11

187
Q

√144

A

12

188
Q

√169

A

13

189
Q

√196

A

14

190
Q

√225

A

15

191
Q

√256

A

16

192
Q

√400

A

20

193
Q

√625

A

25

194
Q

√900

A

30

195
Q

A

1

196
Q

A

8

197
Q

A

27

198
Q

A

64

199
Q

A

125

200
Q

³√1

A

1

201
Q

³√8

A

2

202
Q

³√27

A

3

203
Q

³√64

A

4

204
Q

³√125

A

5

205
Q

PEMDAS

A
Parentheses, 
Exponents,
Multiplication, 
Division, 
Addition, 
Subtraction
206
Q

One of the most common errors involving orders of operations occurs when an expression with multiple terms is ____

A

Subtracted

207
Q

What should you pretend there are around the numerator and denominator of a fraction

A

Parentheses

208
Q

if (a) ÷ (b) = c , Name a, b, and c

A

c then a is the dividend, b the divisor, and c the quotient.