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GMAT > Number Theory > Flashcards

Number Theory Flashcards

Number Theory Memorization

1
Q

First twenty-six prime numbers

A

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 ,41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101

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2
Q

Rules for Prime Numbers

A
  1. There are infinitely many prime numbers
  2. Only positive numbers can be prime
  3. All prime numbers except 2 & 5 end in 1, 3, 7 or 9
  4. All prime numbers aboce 3 are the form of 6n-1 or 6n+1
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3
Q

If A is a factor of B and A is a factor of C then…

A

A is a factor of B+C

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4
Q

If A is a factor of B and B is a factor of C then…

A

A is a factor of C

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5
Q

Find the number of factors for an integer

A
  1. Make prime factorization of integer
  2. n=ap*bq*cr, where a, b, and c are prime factors and p, q, and r are their powers
  3. The number of factors of n = (p+1)(q+1)(r+1)

NOTE: this includes 1 and n itself

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6
Q

Greatest Common Factor (Divisor)

A
  1. List the prime factors of each number.
  2. Multiply those factors both numbers have in common. If there are no common prime factors, the GCF is 1.

Example:

54: 2, 3, 3, 3

36: 2, 2, 3, 3

2*3*3=18

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7
Q

Least Common Multiple

A
  1. Find the prime factors of each number
  2. Take out the factors in the second number that repeat from the first
  3. multiply all remaining factors
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8
Q

mean & median for an evenly spaced set

A

mean=median=(First+Last)/2

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9
Q

Sum of numbers in an evenly spaced set

A

mean of the set x number of items in set

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10
Q

How to count consecutive integers

A

add one before you are done

i.e. How many integers for 6 to 10

10 - 6 = 4 + 1 = 5

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11
Q

How to count consecutive multiples

A

(Last - First) / increment + 1

How many even numbers from 12 to 24

(24 - 12) / 2 + 1 = 7

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12
Q

Sum of Consecutive Integers

A
  1. average the first and last numbers to find precise middle
  2. count number of terms (one then done)
  3. average x number of terms
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13
Q

Products of Consecutive Integers & Divisibility

A

The product of any k consecutive integers is always divisible by k factorial (k!)

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14
Q

Sums of Consecutive Integers and Divisibility

A

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

For any set of consecutive integers with an EVEN number of items, the sums of all the items is NEVER a multiple of the number of items

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15
Q

As numbers between 0 and 1 are raised to a higher exponent, they…

A

deacrease

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16
Q

Exponents with compound bases

A

Exponent may be distributed when multiplying, but not when adding

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17
Q

Adding/Subtracting Exponents

A

IF EXPONENTS HAVE THE SAME BASE:

add them when multiplying

subtract them when dividing

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18
Q

Nested Exponents

A

When raising a power to a power combine exponents by multiplying

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19
Q

When an exponent is negative

A

take the recipricol of the base and make the exponent positive

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20
Q

An Exponent of 0

A

Any non-zero base raised to the power of zero is 1

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21
Q

Fractional Exponents

A

the numerator tells us what power to raise the base to and the denominator tells what root to take

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22
Q

An integer is divisible by 4 if…

A

The integer is divisible by 2, twice

OR

if the last two digits are divisible by by 4

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23
Q

An integer is divisible by 6 if..

A

The integer is divisible by both 2 and 3

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24
Q

An integer is divisible by 8 if…

A

The integer is divisible by 2, three times

OR

if the last three digits are divisible by 8

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25
Q

An integer is divisible by 9 if…

A

the sum of the integers digits is divisible by 9

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26
Q

Find factor pairs for an integer

A
  1. make a table with 2 columns labeled “small” and “larger”
  2. start with 1 in the small column and integer in th large column
  3. Test the next possible factor and repeat until the numbers in the small and large columns run into each other
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27
Q

First and only even prime

A

2

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28
Q

Even +/- Even

A

Even

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29
Q

Odd +/- Odd

A

Even

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30
Q

Odd +/- Even

A

Odd

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31
Q

Odd x Odd

A

Odd

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32
Q

Even x Even

A

Even

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33
Q

Odd x Even

A

Even

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34
Q

Even/Even

A

Even, odd or non integer

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35
Q

Even/Odd

A

Even or non integer

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36
Q

Odd/Even

A

non integer

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37
Q

odd/odd

A

odd or non integer

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38
Q

sum of two prime numbers greater than two will be

A

even,

so, if sum of two unknown prime numbers is odd one of those primes MUST be 2

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39
Q

Consecutive Integers (define)

A

increments of 1

these are also considered consecutive multiples and evenly spaced sets

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40
Q

Consecutive Multiples (define)

A

Multiples of the increment (multiples of 3: 3,6,9,12)

These are also considered evenly spaced sets

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41
Q

Evenly Spaced Sets (define)

A

Constant increments:

example: 2,5,8,11,14

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42
Q

evenly spaced sets (properties)

A

mean and median are equal

mean and median are average of firstand last terms

sum of elements in the set equals:

mean x number of items in set

number of items in set is:

last - first + 1, for consecutive integers

last - first/ increment +1, for consecutive multiples

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43
Q

Factor Foundation Rule as it applies to the product of k consecutive integers

A

product of k consecutive integers is always divisible by k!

any product of 3 consecutive integers will have at least one multiple of 3 and one multiple of 2

any product of 4 consecutive integers will have at least one multiple of 4 and one multiple of 3 and one multiple of 2

etc. etc.

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44
Q

Sums of CONSECUTIVE INTEGERS and divisibility

A

for any set of consecutive integers with an odd number of items, the sum of al the integers is always a multiple of the number of items

for any set of consecutive integers with an even number of items, the sum of all the items is NEVER a multiple of the number of items

SUM= average x number of items

For odd numbers of items the average is an integer and for even numbers it is not

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45
Q

x2

A

x or -x

with a positive exponent we cannot be sure which it is.

we must be told x is positive to affirm that it is.

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46
Q

exponent with a base of 0

A

0

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47
Q

exponent with a base of 1

A

1

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48
Q

x6 = x7 = x15

A

x must equal 0 or 1

(only because of the positive exponent, if they were all negative x could also equal -1)

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49
Q

exponents of a positive proper fraction

A

as the exponent increases the value of the expression decreases

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50
Q

product as compound base with exponent

A

either multiply the numbers in the base or distribute the exponent

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51
Q

sum as a compound base with an exponent

A

you MUST add the numbers together before applying the exponent

(you cannot distribute the exponent)

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52
Q

Exponents: multiplying terms with the same base

A

Add the exponents

34 x 32 = 3(4+2) =36

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53
Q

Exponents: dividing terms with the same base

A

subtract the exponents

36 / 32 = 3(6-2) =34

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54
Q

When raising a power to a power (nested exponents)

A

Combine exponents by multiplying

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55
Q

negative exponent

A

take the reciprocal of the base and change the sign of the exponent to a positive

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56
Q

any number without an exponent…

A

has an implied exponent of 1

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57
Q

any non zero based raised to 0

A

1

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58
Q

fractional exponents

A

numberator tells us what power t oraise the base to and the denominator tells us which root to take

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59
Q

xa * xb

A

xa+b

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60
Q

c3 * c5

A

c8

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61
Q

35 * 38

A

313

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62
Q

5(5n)

A

51(5n)

5n+1

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63
Q

ax * bx

A

(ab)x

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64
Q

24 * 34

A

64

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65
Q

125

A

210 * 35

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66
Q

xa/xb

A

xa-b

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67
Q

25/211

A

1/26

2-6

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68
Q

x10/x3

A

x7

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69
Q

(a/b)x

A

ax/bx

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70
Q

(10/2)6

A

106/26

56

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71
Q

35/95

A

(3/9)5

(1/3)5

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72
Q

(ax)y

A

axy

(ay)x

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73
Q

(32)4

A

32*4

38

34*2

(34)2

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74
Q

x-a

A

1/xa

75
Q

(3/2)-2

A

(2/3)2

4/9

76
Q

2x-4

A

2/x4

77
Q

xa/b

A

b√ xa

(b√x)a

78
Q

274/3

A

³√274

(³√27)4

34

81

79
Q

5√x15

A

X3

80
Q

ax + ax + ax

A

3ax

81
Q

34 + 34 + 34

A

3 * 34

35

82
Q

3x + 3x + 3x

A

3 * 3x

3x+1

83
Q

Simplifying exponential expressions

A

you can simplify exponentialexpressions that are linked by multiplication or division

you CANNOT simplify expresions linked by addition or subtraction (though they can sometimes be factored)

example

74 + 76 = 74(1 + 72)

84
Q

odd roots

A

will have the same sign as the base

85
Q

even roots

A

will only have a positive value

86
Q

√2

A

1.4

87
Q

√3

A

1.7

88
Q

√5

A

2.25

89
Q

√121

A

11

90
Q

112

A

121

91
Q

122

A

144

92
Q

√144

A

12

93
Q

132

A

169

94
Q

√169

A

13

95
Q

142

A

196

96
Q

√196

A

14

97
Q

152

A

225

98
Q

√225

A

15

99
Q

√256

A

16

100
Q

162

A

256

101
Q

202

A

400

102
Q

√400

A

20

103
Q

252

A

625

104
Q

√625

A

25

105
Q

302

A

900

106
Q

√900

A

30

107
Q

23

A

8

108
Q

³√8

A

2

109
Q

33

A

27

110
Q

³√27

A

3

111
Q

43

A

64

112
Q

³√64

A

4

113
Q

53

A

125

114
Q

³√125

A

5

115
Q

n√x / n√y

A

n√x/y

116
Q

n√x * n√y

A

n√xy

117
Q

b√xa

A

(b√x)a

xa/b

118
Q

all perfect squares have an ____ number of total factors

A

odd

119
Q

the prime factorization of a perfect square contains only ____ number of primes

A

even

120
Q

length of a number

A

the number of primes (not necessarily distinct primes)

121
Q

if a prime factorization contains an odd number of primes

A

it is not a perfect square

122
Q

if a number is a perfect cube ten itis fomed from…

A

primes in sets of three

123
Q

if k3 is divisible by 240, what is the least possible value of integer k

A

60

124
Q

N! is a multiple of

A

all integers for 1 to N

125
Q

An = 3n +7

a set of numbers 1 - nth will have what formulas

A

sum = (average) * number in set

      = [(solve for 1) + sequence formula] / 2 \* n

      = [(3\*1 +7) + 3n +7] /2 \* n

      = (10 + 3n + 7)/2 \* n

      = solve
126
Q

range of a set

A

first number - last number

127
Q

sum of n consecutive integers

when n is odd

A

is divisible by n

128
Q

sum of n consecutive integers

when n is even

A

is NOT divisible by N

129
Q

the sign of an integer is unclear when….

keep this in mind on data sufficiency problems!!!!

A

it is raised to a positive power

absolute value of integer

is ab< 0 ?

|a| * b <0

(this means b is negative but since we do not know if a is negative, we cannot answer the question)

a4 * b < 0

(this means b is negative but the positve power hides te sign of a, if a is negative, then ab is positive, is ab>0 positive, then ab <0)

130
Q

if a/b yields a remainder of 5 & c/d yields a remainder of 8, and a, b, c, and d are integers, what is the smallest possible value of b+d

A

remainders must be smaller than the divisor

5<b 8<d

6+9 = 15

131
Q

if x leaves a remainder of 4 after division by 7 and y leaves a remainder of 2 after divsion by 7 what is the remainder of x+y/7

A

6

you can add and subtract remainders, directly, as long as you correct excess or negative remainders

132
Q

if x leaves a remainder of 4 after division by 7 and z leaves a remainder of 5 after division by 7 then what is the remainder of x+z/7

A

you can add and subtract remainders as long as you correct excess or negative remainders

4+5=9 - 7 = 2

133
Q

if x leaves a remainder of 4 after division by 7 and z leaves a remainder of 5 after division by 7 then what is the remainder of x-z/7

A

you can add and subtract remainders as long as you correct excess or negative remainders

4-5 = -1 + 7 = 6

134
Q

if x has a remainder of 4 when divided by 7 and z has a remainder of 5 when divided by 7 what is the remainder when x*z is divided by 7

A

you can multiply remainders as long as you correct excess remainders at the end

4*5 = 20 -(7*2)= 6

135
Q

remainders and decimals

how to convert the decimal portion into a remainder

A

x (remainder) /divisor=decimal

example: 17/5=3.4

.4 = x/5

136
Q

when add or subtracting two numbers, neither of which is divisible by 2… the result will

A

always be divisible by 2

137
Q

write an arbitrary odd integer algebraically

A

2n + 1

138
Q

absolute value | x- y |, define

A

distance between x and y

139
Q

x3 - x is the same as

and if

x3 - x = p and x is odd, and x >1, is p divsible by 24

A

x(x2 - 1)

x(x - 1)(x +1)

consecutive integers

since x is odd, x-1 and x+1 are even

each divisible by 2 one at least one divisible by 4,

4x2=8

in a set of 3 consecutive integers at least one will be divisible by 3

3x8=24, YES

140
Q

x2 - x

A

x(x-1)

141
Q

x4 - x2

A

x2(x2-1)

x2(x+1)(x-1)

142
Q

75 - 73

A

73(72 - 1)

48 * 73

143
Q

58 + 59 +510

A

58(1+5+52)

31 * 58

144
Q

z3 - z

A

z(z2-1)

z(z+1)(z-1)

145
Q

10(b+1)

A

10(10b)

146
Q

10(b-1)

A

10b/10

147
Q

35 + 35 + 35

A

3(35)

36

148
Q

ab - ab-1

A

ab(1 - a-1)

ab-1(a - 1)

149
Q

pq + pr + qs +rs

A

p(q + r) + s(q+r)

(p+s)(q+r)

150
Q

adding or subtracting roots

A

roots act like variables, you can only combine them if they are like terms

simplify roots

√80 - √45

4√5 - 3√5 = √5

151
Q

how to rationalize a denominator with square roots

A

use conjugates to cancel out the square root

152
Q

the sum of two multiples of a number

A

is a multiple of that number

example :

a multiple of 3 + another multiple of 3 = multiple of 3

12 + 9 = 21

153
Q

How to determine if a r/s will result in a terminating decimal

A
  1. you must know s
  2. the remainder of r/s must be less than s
  3. put all possible remainders of s over s and find out if any of those are terminating decimals

example s=4

only possible remainders are 1, 2 and 3

1/4 = .25, 2/4 = .5, 3/4 = .75 – so yes, r/s in this case will have a terminating decimal

154
Q

the sum of all distinct factors of a perfect square is always …

A

odd

example

4: 2 + 4 + 1 = 7
9: 3 + 9 + 1 = 13

155
Q

convert to decimal and percent

1/2

A

.5

50%

156
Q

convert to decimal and percent

1/3

A

.33

33%

157
Q

convert to decimal and percent

2/3

A

.66

66%

158
Q

convert to decimal and percent

4/6

A

.66

66.6%

159
Q

convert to decimal and percent

2/6

A

.33

33%

160
Q

convert to decimal and percent

3/6

A

.5

50%

161
Q

convert to decimal and percent

4/6

A

.66

66.6%

162
Q

convert to decimal and percent

5/6

A

.83

83%

163
Q

convert to decimal and percent

1/4

A

.25

25%

164
Q

convert to decimal and percent

2/4

A

.5

50%

165
Q

convert to decimal and percent

3/4

A

.75

75%

166
Q

convert to decimal and percent

1/5

A

.2

20%

167
Q

convert to decimal and percent

2/5

A

.4

40%

168
Q

convert to decimal and percent

3/5

A

.6

60%

169
Q

convert to decimal and percent

4/5

A

.8

80%

170
Q

convert to decimal and percent

1/8

A

.125

12.5%

171
Q

convert to decimal and percent

2/8

A

.25

25%

172
Q

convert to decimal and percent

3/8

A

.375

37.5%

173
Q

convert to decimal and percent

4/8

A

.5

50%

174
Q

convert to decimal and percent

5/8

A

.625

62.5%

175
Q

convert to decimal and percent

6/8

A

.75

75%

176
Q

convert to decimal and percent

7/8

A

.875

87.5%

177
Q

182

A

324

178
Q

192

A

361

179
Q

202

A

400

180
Q

22

23

24

25

26

27

28

29

210

A

4

8

16

32

64

128

256

512

1024

181
Q

32

33

34

A

9

27

81

182
Q

42

43

44

A

16

64

256

183
Q

52

53

54

A

25

125

625

184
Q

172

A

289

GMAT (7 decks)

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