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GMAT > Arithmetic > Flashcards

Arithmetic Flashcards

1
Q

Definition of an even integer

A

an integer that is divisible by 2, written as 2n

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2
Q

Definition of an odd integer

A

any integer that is not divisible by 2, written as 2n+1

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3
Q

Definition of a prime number

A

a positive integer that has exactly two different positive divisors (factors): 1 and itself (2,3,5,7,11,13..)

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4
Q

Is 1 a prime number?

A

no since it only has one divisor

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5
Q

Definition of integer

A

any whole negative or positive number, including 0 (that they are not fractions.)

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6
Q

Consecutive even integers can be represented by

A

2n (n is an integer)

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7
Q

Consecutive odd integers can be represented by

A

2n +1 (n is an integer)

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8
Q

Is 0 an odd or an even number?

A

0 is an even number

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9
Q

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

A

a is factor of b + c

a is factor of b - c

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10
Q

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

A

a is factor of c

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11
Q

Definition of arithmetic mean

A

the average of n numbers in a set of data

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12
Q

Define the median

A

the middle number of n numbers (rank from small to large) in a set of data

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13
Q

Define the mode

A

the number that appears the most in a set of data

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14
Q

How do we measure the degree of a set of number spread out/ disperse?

A

calculate the range or standard deviation

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15
Q

how to calculate the range

A

subtract the greatest value to the least value in set of n numbers

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16
Q

how to calculate the standard deviation

A

(1) find the arithmetic mean, (2) find the differences between the mean and each of the n numbers, (3) square each of the differences, (4) find the average of the squared differences, and (5) take the nonnegative square root of this average.

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17
Q

The greater the standard deviation, the ________

A

the more the data spread away from the mean

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18
Q

If all the elements of a set A are also elements of a set B, then

A

A is the subset of B

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19
Q

For any two sets A and B, the union of A and B is

A

the set of all elements that are in A or in B or in both.

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20
Q

The union is denoted by

A

A ∪ B

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21
Q

The intersection of A and B is

A

the set of all elements that are in both A and B.

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22
Q

the intersection is denoted

A

A ∩ B

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23
Q

Two sets that have no elements in common are said to be

A

disjoint or mutually exclusive

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24
Q

If two sets A and B are NOT disjoint/ mutually exclusive, then |A ∪ B | is

A

A| + | B | − |A ∩ B |

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25
Q

If A and B are disjoint/ mutually exclusive then

A

|A ∪ B | = | A | + | B |

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26
Q

If an object is to be chosen from a set of m objects and a second object is to be chosen from a different set of n objects, how many ways of choosing both objects simultaneously?

A

m x n ways

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27
Q

The number of ways of ordering the letters A, B, and C?

A

3! = 6

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28
Q

If the order of selection is not relevant and only k objects are to be selected from a larger set of n objects, then the number of combination is ____

A

nPk = n! : {k! (n - k)!}

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29
Q

For experiments in which all the individual outcomes are EQUALLY LIKELY, the probability of an event E is

A

the number of outcomes in E divide by total number of possible outcomes

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30
Q

If the event “A and B” is impossible (that is, A ∩ B has no outcomes), then

A

A and B are said to be mutually exclusive

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31
Q

If A and B are mutually exclusive events, then P (A and B) and P (A or B)

A

P (A and B ) = 0

P (A or B) = P(A) + P(B)

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32
Q

Two events A and B are said to be independent if

A

the outcome of one event does not influence or affect the outcome of the other event

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33
Q

If any independent events A and B occur, then P (A and B) and P (A or B)

A

P(A and B) = P(A) x P(B)

P(A) + P(B) - P(A and B)

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34
Q

Assign word to each symbol:
P (A ∩ B )
P (A ∪ B)

A

P (A and B) - intersection of A and B

P (A or B) - union of A and B

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35
Q

T or F. If P (A) + P (B) is greater than 1 then A and B are not mutually exclusive

A

True. Because probability can’t be greater than 1 so there exists P (A and B)

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36
Q

even +/- even =?

A

even

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37
Q

even +/- odd = ?

A

odd

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38
Q

odd +/- odd = ?

A

even

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39
Q

even * even = ?

A

even

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40
Q

even * odd = ?

A

even

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41
Q

odd * odd =?

A

odd

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42
Q

What is the smallest prime number?

A

2

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43
Q

What is the general formula for prime number?

A

6n +1 or 6n-1 (except for 2 and 5)

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44
Q

An integer is divisible by 3 if

A

the SUM of the integer’s DIGITS is divisible by 3

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45
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 4

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46
Q

An integer is divisible by 5 if

A

the integer ends in 0 or 5

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47
Q

An integer is divisible by 6 if

A

the integer is divisible by BOTH 3 and 2

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48
Q

An integer is divisible by 7 if

A

you take the LAST digit, DOUBLE it, and SUBTRACT it from the rest of the number, if the answer is divisible by 7 ( including 0) then the number is divisible by 7

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49
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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50
Q

An integer is divisible by 9 if

A

the SUM of the integer’s DIGITS is divisible by 9

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51
Q

An integer is divisible by 10 if

A

the integer ends in 0

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52
Q

An integer is divisible by 11 if

A

you SUM every SECOND digit and then SUBSTRACT all other digits and the answer is 0 or divisible by 11, then the number is divisible by 11

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53
Q

An integer is divisible by 12 if

A

the integer is divisible by BOTH 3 and 4

54
Q

An integer is divisible by 25 if

A

the integers ends in 00, 25, 50, 75

55
Q

How do you express 15 is divided by 6 in math?

A

15/6

56
Q

If x and y are positive integers, there exist unique integers q (the quotient) and r (remainder) such that

A

y = x (divisor) . quotient + remainder

57
Q

T or F. If you have a divisor throughout, you can not add and subtract remainders directly, as long as you correct excess or negative remainders

A

False. For example:
If x leaves a remainder of 4 after division by 7 and z leaves a remainder of 5 after division by 7. Algebraically,
x = 7q1 + 4
z = 7q2 + 5
- x + z = 7 (q1+q2) + 9 = 7 (q1+q2) + 7 +2 so x+ z is multiple of 7, plus 2 (remainder)
- x - z = 7 (q1 - q2) + 4 -5 = 7 (q1 - q2 - 1) - 1 = 7 (q1 - q2 - 1) - 7+6 so x-z is multiple of 7, plus 6 (remainder)

58
Q

T or F. If you have a divisor throughout, you can multiply remainders directly, as long as you correct excess or negative remainders

A

True. With x, z divided by 7 example, remainder (4). remainder(5) = remainder (20) . Taking out excess of 7 twice, we have 6 left. Thus, remainder of x.z is 6

59
Q

If any number with ones digit equal to 0, then that number is divisible by?

A

10 or 5

60
Q

if the sum of the digits of x is equal to 21, you can infer that x is divisible by?

A

by 3 but not 9

61
Q

T or F. Saying 3 is a divisor of 12 is the same as 3 is factor of 12

A

True

62
Q

How do you write 3 divides 12? How else this can be written?

A

12/3 or 12 is divided by 3

63
Q

How do you express “ m is a multiple of n”

A

m = k.n. For example

12 is a multiple of 3

64
Q

T or F. An integer can have more factors than its multiple?

A

False.

8 only has four factors while multiple of 8 is unlimited

65
Q

If you add/subtract a multiple of N to a non-multiple of N, the result is

A

non- multiple of N

66
Q

If you add/subtract two non-multiple of N, the result is

A

could be either a multiple of N or a non-multiple of N

67
Q

How do you find the greatest common factor (divisor)?

A

1) Prime- factorization
2) Of all integers, Multiply the common factors that have the lowest power
Ex: GCF of 120 and 100 is 20

68
Q

How do you find the lowest (least) common multiple?

A

1) Prime- factorization
2) Multiply all prime factors but only choose the highest power of common factors
Ex: GCF of 120 and 100 is 600

69
Q

Where do you often need to do calculation of LCM?

A

find the common multiple of two fractions’ denominator

70
Q

What is unconventional way to find LCM (a,b) ?

A

(a x b)/ GCF (a,b)

71
Q

A number is a perfect square if___ or ____?

A

1) The number of total different factors is an odd number

2) The exponents of all unique prime factors are multiple of two

72
Q

Is 132,300 a perfect square?

A

No because: 132,300 = 2^2.3^3. 5^2. 7^2

1) 3 is raised to an odd power
2) its total number of factors is 108 (not odd number)

73
Q

If a set contains consecutive integers (odd/even or evenly spaced), then the mean is ?

A

mean = median = (first number + last number) /2

74
Q

What is the sum of n consecutive integers?

A

(mean of n) x n

75
Q

The Σ of n consecutive integers is always divisible by n, when?

A

n is odd

76
Q

T or F. When n consecutive integers is an odd number, the average is never an integer

A

F.

Since the sum is divisible by n, its mean is an integer

77
Q

the Σ of n consecutive integers is never divisible by n, when?

A

n is even

78
Q

T or F. When n consecutive integers is an even number, the average is an integer

A

F.

Since the sum is never divisible by n, its mean is never an integer

79
Q

T or F. If a mean of n consecutive integers is an integer, n is an odd number

A

T.

80
Q

What does consecutive multiples set mean?

A

each element in the set is the result of increment of multiples
Ex: {12,16,20,24} is a set of consecutive multiple of 4

81
Q

How do you express an equation in evenly spaced integers (6, 11, 16, 21)?

A

a(n) = a(1) + d (n-1)

82
Q

T or F. In evenly spaced integers, if n number of elements is even, the sum is not divisible by n

A

T. This characteristic applies insofar to evenly spaced consecutive integers

83
Q

In consecutive integers set n, the product of set is always divisible by

A

factorial of n (n!)

84
Q

T or F. Despite the total number of elements in evenly space set is an even/odd number, the mean is equal to median

A

T.

85
Q

7,654.321 List the position of each digit in English term

A 
7 = thousands
6= hundreds
5 = tens
4 = ones of units
3 = tenths
2= hundredths
1= thousandths
86
Q

What is the shortcut for decimal places in square/ cube?

Ex: (0.2) ^ 6 = ?

A

The number of decimal places in the result of a square/ root decimal is 2/3 TIMES the number of decimal places in the original decimal

2^6 = 32 move to left six times -> 0.000032

87
Q

Can you apply decimal places short cut to convert 0.00005 to the value before its squared/cubed?

A

No because the # of decimal places is not divisible by 2 or 3 (exponent values) or other integer exponent value

88
Q

what are the results of:

  • 10% greater than original
  • 75% of the original
A

110% of the original

25% less than

89
Q

what number is 10% greater than 60?

A

110% . 60 = 66

90
Q

How do you convert 17/25 to percentage?

A

Multiply by 4 for numerator and denominator. We have 68/100 = 68%

91
Q

In combinatorics, “or” mean

“and” mean

A

“add”

“multiply”

92
Q

In combinatorics, with replacement, there is how many to arrange n distinct objects?

A

there is n! to arrange distinct objects

93
Q

If there are two groups with one has k members, how many arrangements are there?

A

n! / k! (n-k)!

94
Q

In combinatorics, without replacement implying….

A

you can’t put the objects back into the pool

95
Q

The total probability of n independent events is

A

the product of all probabilities of those independent events

96
Q

if two events are mutually exclusive, then probability of two events is

A

the sum of those two probabilities of events

97
Q

What happen to the SD if we decrease/ increase in all elements of a set by a constant percentage/factor?

A

the SD will also increase/ decrease by the same percentage/factor

98
Q

T or F. Increase/ decrease in all elements by a constant value will increase/ decrease SD

A

False

99
Q

If there is a new element added to a set then

  • new standard deviation is greater than original if
  • new SD = old SD if
  • new SD < old SD if
A
  • absolute value of the difference between new value and mean greater than 0
  • absolute value of the difference between new value and mean equal 0
  • absolute value of the difference between new value and mean less than 0
100
Q

11.1 % equal to fraction of

A

1/9

101
Q

12.5% equal to fraction of

A

1/8

102
Q

16.7% equal to fraction of

A

1/6

103
Q

83.3% equal to fraction of

A

5/6

104
Q

125% equal to fraction of

A

5/4

105
Q

133% equal to fraction of

A

4/3

106
Q

175% equal to fraction of

A

7/4

107
Q

What is a smart number you should pick in GMAT?

A

100

108
Q

T or F. You can only pick one smart number for one variable.

A

T.

109
Q

What does prime factorization in the fraction denominator (in fully reduced form) must consist in order to have a fraction as a terminate decimal?

A

the prime factorization must consist of only 2’s or only 5 or only 2 and 5

Ex: 3/105 doesn’t have terminate decimal

110
Q

T or F. There is only one mode in a set of data

A

F.

There can be as many modes in a set of data

111
Q

What does the frequency of a number in a set of data tell us?

A

the number of times does that number occur in a set of data.

112
Q

T or F. An odd number divided by any other integer cannot produce an even integer

A

T

113
Q

T or F. An odd number divided by an even number cannot produce an integer

A

T

Odd number doesn’t have factor of 2

114
Q

Consecutive multiples of n have a GCF of ___ ?

A

n

Ex: n = 2 -> numbers: 2,4, 6 all have GCF is 2

115
Q

How to calculate the total prime factors (length)?

A

add the value of exponents of all prime factors

116
Q

A number is a perfect cubes if___ or ____? (pg.97 Manhattan)

A

1) the number of total factors is odd number

2) all the powers of primes are multiple of 3 in the factorization

117
Q

T or F. The remainder must be smaller than the divisor

A

T

118
Q

What are the possible remainders when divide an integer by a positive integer N? (p.100 Manhattan- Number Properties)

A

0 -> N - 1

Ex: If N =4, there are 4 remainders (0,1,2,3)

119
Q

If M has prime factorization a^x . b^y . c^z , what is the number of different factors of M?

A

(x+1). (y+1). (z+1)

120
Q

Compound Interest Formula:

A

C = P (1 +r/n) ^ n.t

P = Principal
r= interest rate
n = number of times per year
t = number of years
121
Q

T or F. For some compound interest problems on GMAT, it may be easier finding the solution without the formula

A

True.

122
Q

T or F. For ratio problems, you can pick ‘x’ as unknown number to solve for each individual variables (p.66 Manhattan- F,D and %)

A

True.

123
Q

What are the relative values on GMAT?

A

fractions, decimals, percents or ratios

124
Q

What are the concrete values on GMAT?

A

specific number of tickets sold, liters of water, etc

125
Q

T or F. If a data sufficiency questions ask for the relative value of two pieces of a ratio, any statement that gives the relative value of ANY two pieces of the ratio will be sufficient

A

T.

Pg. 93 (Manhattan - FDPR)

126
Q

Composite number

A

Number that has more than two factors. Therefore, composite number is non-prime number

127
Q

Is 0 a composite number?

A

NO

128
Q

T or F: For positive numbers, if the starting faction is less than 1, the fraction gets farther from 1 (but still <1) as you add the same number to the top and bottom

A

F
the fraction get closer to 1

129
Q

How does a faction (that is less than 1) change when you add the same number to top and bottom?

A

It increase
E.g: 1/2 < 2/3

130
Q

How does a faction (that is greater than 1) change when you add the same number to top and bottom?

A

It decrease
E.g: 3/2 > 4/3

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