Arithmetic Flashcards
Definition of an even integer
an integer that is divisible by 2, written as 2n
Definition of an odd integer
any integer that is not divisible by 2, written as 2n+1
Definition of a prime number
a positive integer that has exactly two different positive divisors (factors): 1 and itself (2,3,5,7,11,13..)
Is 1 a prime number?
no since it only has one divisor
Definition of integer
any whole negative or positive number, including 0 (that they are not fractions.)
Consecutive even integers can be represented by
2n (n is an integer)
Consecutive odd integers can be represented by
2n +1 (n is an integer)
Is 0 an odd or an even number?
0 is an even number
If a is factor b and a is factor of c, then
a is factor of b + c
a is factor of b - c
If a is factor of b and b is factor of c, then
a is factor of c
Definition of arithmetic mean
the average of n numbers in a set of data
Define the median
the middle number of n numbers (rank from small to large) in a set of data
Define the mode
the number that appears the most in a set of data
How do we measure the degree of a set of number spread out/ disperse?
calculate the range or standard deviation
how to calculate the range
subtract the greatest value to the least value in set of n numbers
how to calculate the standard deviation
(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.
The greater the standard deviation, the ________
the more the data spread away from the mean
If all the elements of a set A are also elements of a set B, then
A is the subset of B
For any two sets A and B, the union of A and B is
the set of all elements that are in A or in B or in both.
The union is denoted by
A ∪ B
The intersection of A and B is
the set of all elements that are in both A and B.
the intersection is denoted
A ∩ B
Two sets that have no elements in common are said to be
disjoint or mutually exclusive
If two sets A and B are NOT disjoint/ mutually exclusive, then |A ∪ B | is
A| + | B | − |A ∩ B |
If A and B are disjoint/ mutually exclusive then
|A ∪ B | = | A | + | B |
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?
m x n ways
The number of ways of ordering the letters A, B, and C?
3! = 6
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 ____
nPk = n! : {k! (n - k)!}
For experiments in which all the individual outcomes are EQUALLY LIKELY, the probability of an event E is
the number of outcomes in E divide by total number of possible outcomes
If the event “A and B” is impossible (that is, A ∩ B has no outcomes), then
A and B are said to be mutually exclusive
If A and B are mutually exclusive events, then P (A and B) and P (A or B)
P (A and B ) = 0
P (A or B) = P(A) + P(B)
Two events A and B are said to be independent if
the outcome of one event does not influence or affect the outcome of the other event
If any independent events A and B occur, then P (A and B) and P (A or B)
P(A and B) = P(A) x P(B)
P(A) + P(B) - P(A and B)
Assign word to each symbol:
P (A ∩ B )
P (A ∪ B)
P (A and B) - intersection of A and B
P (A or B) - union of A and B
T or F. If P (A) + P (B) is greater than 1 then A and B are not mutually exclusive
True. Because probability can’t be greater than 1 so there exists P (A and B)
even +/- even =?
even
even +/- odd = ?
odd
odd +/- odd = ?
even
even * even = ?
even
even * odd = ?
even
odd * odd =?
odd
What is the smallest prime number?
2
What is the general formula for prime number?
6n +1 or 6n-1 (except for 2 and 5)
An integer is divisible by 3 if
the SUM of the integer’s DIGITS is divisible by 3
An integer is divisible by 4 if
the integer is divisible by 2 TWICE, or if the LAST TWO digits are divisible by 4
An integer is divisible by 5 if
the integer ends in 0 or 5
An integer is divisible by 6 if
the integer is divisible by BOTH 3 and 2
An integer is divisible by 7 if
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
An integer is divisible by 8 if
the integer is divisible by 2 THREE TIMES, or if the LAST THREE digits are divisible by 8
An integer is divisible by 9 if
the SUM of the integer’s DIGITS is divisible by 9
An integer is divisible by 10 if
the integer ends in 0
An integer is divisible by 11 if
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
An integer is divisible by 12 if
the integer is divisible by BOTH 3 and 4
An integer is divisible by 25 if
the integers ends in 00, 25, 50, 75
How do you express 15 is divided by 6 in math?
15/6
If x and y are positive integers, there exist unique integers q (the quotient) and r (remainder) such that
y = x (divisor) . quotient + remainder
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
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)
T or F. If you have a divisor throughout, you can multiply remainders directly, as long as you correct excess or negative remainders
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
If any number with ones digit equal to 0, then that number is divisible by?
10 or 5
if the sum of the digits of x is equal to 21, you can infer that x is divisible by?
by 3 but not 9
T or F. Saying 3 is a divisor of 12 is the same as 3 is factor of 12
True
How do you write 3 divides 12? How else this can be written?
12/3 or 12 is divided by 3
How do you express “ m is a multiple of n”
m = k.n. For example
12 is a multiple of 3
T or F. An integer can have more factors than its multiple?
False.
8 only has four factors while multiple of 8 is unlimited
If you add/subtract a multiple of N to a non-multiple of N, the result is
non- multiple of N
If you add/subtract two non-multiple of N, the result is
could be either a multiple of N or a non-multiple of N
How do you find the greatest common factor (divisor)?
1) Prime- factorization
2) Of all integers, Multiply the common factors that have the lowest power
Ex: GCF of 120 and 100 is 20
How do you find the lowest (least) common multiple?
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
Where do you often need to do calculation of LCM?
find the common multiple of two fractions’ denominator
What is unconventional way to find LCM (a,b) ?
(a x b)/ GCF (a,b)
A number is a perfect square if___ or ____?
1) The number of total different factors is an odd number
2) The exponents of all unique prime factors are multiple of two
Is 132,300 a perfect square?
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)
If a set contains consecutive integers (odd/even or evenly spaced), then the mean is ?
mean = median = (first number + last number) /2
What is the sum of n consecutive integers?
(mean of n) x n
The Σ of n consecutive integers is always divisible by n, when?
n is odd
T or F. When n consecutive integers is an odd number, the average is never an integer
F.
Since the sum is divisible by n, its mean is an integer
the Σ of n consecutive integers is never divisible by n, when?
n is even
T or F. When n consecutive integers is an even number, the average is an integer
F.
Since the sum is never divisible by n, its mean is never an integer
T or F. If a mean of n consecutive integers is an integer, n is an odd number
T.
What does consecutive multiples set mean?
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
How do you express an equation in evenly spaced integers (6, 11, 16, 21)?
a(n) = a(1) + d (n-1)
T or F. In evenly spaced integers, if n number of elements is even, the sum is not divisible by n
T. This characteristic applies insofar to evenly spaced consecutive integers
In consecutive integers set n, the product of set is always divisible by
factorial of n (n!)
T or F. Despite the total number of elements in evenly space set is an even/odd number, the mean is equal to median
T.
7,654.321 List the position of each digit in English term
7 = thousands 6= hundreds 5 = tens 4 = ones of units 3 = tenths 2= hundredths 1= thousandths
What is the shortcut for decimal places in square/ cube?
Ex: (0.2) ^ 6 = ?
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
Can you apply decimal places short cut to convert 0.00005 to the value before its squared/cubed?
No because the # of decimal places is not divisible by 2 or 3 (exponent values) or other integer exponent value
what are the results of:
- 10% greater than original
- 75% of the original
110% of the original
25% less than
what number is 10% greater than 60?
110% . 60 = 66
How do you convert 17/25 to percentage?
Multiply by 4 for numerator and denominator. We have 68/100 = 68%
In combinatorics, “or” mean
“and” mean
“add”
“multiply”
In combinatorics, with replacement, there is how many to arrange n distinct objects?
there is n! to arrange distinct objects
If there are two groups with one has k members, how many arrangements are there?
n! / k! (n-k)!
In combinatorics, without replacement implying….
you can’t put the objects back into the pool
The total probability of n independent events is
the product of all probabilities of those independent events
if two events are mutually exclusive, then probability of two events is
the sum of those two probabilities of events
What happen to the SD if we decrease/ increase in all elements of a set by a constant percentage/factor?
the SD will also increase/ decrease by the same percentage/factor
T or F. Increase/ decrease in all elements by a constant value will increase/ decrease SD
False
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
- 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
11.1 % equal to fraction of
1/9
12.5% equal to fraction of
1/8
16.7% equal to fraction of
1/6
83.3% equal to fraction of
5/6
125% equal to fraction of
5/4
133% equal to fraction of
4/3
175% equal to fraction of
7/4
What is a smart number you should pick in GMAT?
100
T or F. You can only pick one smart number for one variable.
T.
What does prime factorization in the fraction denominator (in fully reduced form) must consist in order to have a fraction as a terminate decimal?
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
T or F. There is only one mode in a set of data
F.
There can be as many modes in a set of data
What does the frequency of a number in a set of data tell us?
the number of times does that number occur in a set of data.
T or F. An odd number divided by any other integer cannot produce an even integer
T
T or F. An odd number divided by an even number cannot produce an integer
T
Odd number doesn’t have factor of 2
Consecutive multiples of n have a GCF of ___ ?
n
Ex: n = 2 -> numbers: 2,4, 6 all have GCF is 2
How to calculate the total prime factors (length)?
add the value of exponents of all prime factors
A number is a perfect cubes if___ or ____? (pg.97 Manhattan)
1) the number of total factors is odd number
2) all the powers of primes are multiple of 3 in the factorization
T or F. The remainder must be smaller than the divisor
T
What are the possible remainders when divide an integer by a positive integer N? (p.100 Manhattan- Number Properties)
0 -> N - 1
Ex: If N =4, there are 4 remainders (0,1,2,3)
If M has prime factorization a^x . b^y . c^z , what is the number of different factors of M?
(x+1). (y+1). (z+1)
Compound Interest Formula:
C = P (1 +r/n) ^ n.t
P = Principal r= interest rate n = number of times per year t = number of years
T or F. For some compound interest problems on GMAT, it may be easier finding the solution without the formula
True.
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 %)
True.
What are the relative values on GMAT?
fractions, decimals, percents or ratios
What are the concrete values on GMAT?
specific number of tickets sold, liters of water, etc
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
T.
Pg. 93 (Manhattan - FDPR)
Composite number
Number that has more than two factors. Therefore, composite number is non-prime number
Is 0 a composite number?
NO
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
F
the fraction get closer to 1
How does a faction (that is less than 1) change when you add the same number to top and bottom?
It increase
E.g: 1/2 < 2/3
How does a faction (that is greater than 1) change when you add the same number to top and bottom?
It decrease
E.g: 3/2 > 4/3
