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GMAT Math Review > Arithmetic > Flashcards

Flashcards in Arithmetic Deck (58):

What is an integer?

A whole number that can be positive, negative or zero.

Ex: 0 , 1 , -1 NOT 1.24, -1.24


What is a divisor?

A number by which another number is to be divided.

Ex: If x and y are integers and x is not equal to zero x is a divisor (factor) of y provided that y = xn for some integer n.


What is a dividend?

A number to be divided by another number (divisor).

Ex: If x / y then x is a dividend to which y is the divisor.


What is a multiple?

A multiple is the resulting number of two integers being multiplied.

Ex: If z is a multiple of xn for some integer n and some integer x then x and n are both factors of z and z is divisible by or or n.


What are quotients and remainders?

Quotient is the result obtained after a dividend interacts with a divisor. The remainder is what is left over.

Ex: y = xq + r ; r is greater than or equal to zero and less than x. Where x is the divisor.

Ex: 28 / 8 ; q = 3 and remainder is equal to 4


What are odd and even integers?

Any integer that is divisible by 2 is an even integer ( can be negative or positive ), zero is even!

Any integer that is not divisible by 2 is an odd integer ( can be negative or positive ).


What are the rules of odd and even integers?

*If at least one factor of a product of integers is even then the product is even.

*If all factors of a product of integers is odd then the product is odd.

*If two integers are both even or both odd, their sum and their difference are even. Otherwise, their sum and their difference are odd.

Ex: -8 is even and 29 is odd


What are prime numbers?

A prime number is a positive integer that has exactly two different positive divisors, 1 and itself. The number 1 is not a prime number! Every integer greater than 1 is either prime or can be uniquely expressed as a product of prime factors.


What are the prime numbers from 1 to 100?

There are 25 prime numbers between 1 and 100. They are 2, 3, 5, 7, 11, 13, 17, 19,23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, and 97.

2 is the only number that is even and prime.


What are the formulas for consecutive integers, consecutive even integers, and consecutive odd integers?

n + 1, n + 2, n + 3, ...,

2n, 2n + 2, 2n +4, …,

2n + 1, 2n + 3, 2n + 5,...,


Name two formulas to express 1

n / n = 1
n * 1 / n = 1
Where n is not equal to zero


Define numerator and denominator.

n/d such that d can never be zero.


What constitutes two fractions being equivalent?

Two fractions are equivalent if they represent the same number.

Ex: 8/36 and 14/63 both represent the number 2/9.


What is the GCD?

A fraction can be reduced to its lowest terms by dividing by the greatest common divisor ( or factor ) of the numerator and denominator.


How do you add and subtract fractions?

If they have the same denominator then perform the desired function on the numerator.

If they do not have the same denominator then try and locate the least common multiple to convert them to similar fractions.


How do you multiply fractions?

To multiply two fractions, simply multiply the two numerators and the two denominators.


How do you divide fractions?

To divide by a fraction invert the divisor and multiply.


How do you convert a mixed number to a fraction?

A mixed number is a number that consists of a whole number and a fraction. To convert simply multiply the whole number by the denominator and add the result to the numerator.


Converting fractions to decimals.

321 / 1,000 = .321
321 / 10,000 = .0321
156 / 100 = 1.56


Recap scientific notation.

x 10 ^ 2 move the decimal to the right.

x 10 ^ -2 move the decimal to the left.


Adding and subtracting decimals.

Simply remember to match the decimal points up. Add zeros if necessary.


Multiplication of decimals.

Simply multiply the numbers as normal and move decimals over in the product equal to number of digits to the right of decimals in integers.


Dividing by decimals.

To divide a number (the dividend) by a decimal (the divisor), move the decimal point of the divisor to the right until the divisor is a whole number. Then move the decimal point of the dividend the same number of places to the right.


What are real numbers?

All real numbers correspond to points on the number line and all points on the number line correspond to real numbers.


Define absolute value.

The distance between a number and zero on the number line. The value of any nonzero number is positive.


Some properties of real numbers include.

a. x + y = y + x AND xy = yx
b. (x + y) + z = x + (y + z) AND (xy)z = x(yz)
c. x(y + z) = xy + xz
d. If x and y are both positive, then x + y and xy are positive
e. If x and y are both negative, then x + y is negative and xy is positive.
f. If x is positive and y is negative, then xy is negative.
g. |x + y| less than or equal to |x| + |y|


Ratio and Proportion

The ratio of a to b is defined as a/b where b does not equal zero.


Define proportion.

Is a statement that two ratios are equal.

Ratios are proportional if they divide into the same number.


How can you find a percent of a certain number?

To find a certain percent of a number, multiply the number by the percent expressed as a decimal or fraction.


Percents greater than 100%

Are represented by numbers greater than 1

Ex: 250% = 2.5


Percents less than 1%

.5% is equal to .005


Percent change

Calculate the increase or decrease and divide by the original amount.

Note that percent increase and percent decrease will not be the same for any two x and y.


Percent increase greater than 100% - If the cost of a certain house in 1983 was 300 percent of its cost in 1970, by what percent did the cost increase?

3n – n / n = 2 or 200%


For k^n if k is equal to 2, what are the resulting products of n defined as?

The products are powers of 2


Define an nth root of a number k.

An nth root of a number k is a number that, when raised to the nth power, is equal to k. 2^4 = 16 , therefore 2 is a 4th root of 16.


Some properties of squaring numbers.

Squaring a number that is greater than 1 results in a larger number

Squaring a number between 0 and 1 results in a smaller number.


Some properties of square roots.

A square root of a number n is a number that, when squared, is equal to n.

The square root of a negative number is not a real number.

Every positive number n has two square roots, one positive and the other negative.


Some properties of cube roots.

Every real number r has exactly one real cube root.
8 = 2
-8 = -2


What are some properties of a median?

It can be less than, equal to, or greater than the mean.


Define mode.

The number that occurs the most frequently, there can be two modes.


Define range.

Greatest value minus minimum value.


Properties and calculation of Standard Deviation.

(1) Calculate the mean. (2) Calculate n – mean for each number. (3) Square each of the differences. (4) Calculate the average of the squared differences. (5) Take the nonnegative square root of this number.

The smaller the STD the closer the numbers are dispersed around the mean.


What is a frequency distribution?

It displays the number of times each number occurs in a given set.

Basic descriptive statistics should be derived from such a distribution.


Define set.

Collection of numbers or other objects. Order does not matter. A finite set is denoted as |S| followed by number of elements in the set.

If all elements of a set exist in another set it is a subset.



of two sets is equal to all elements that are in A or B or in both ( think full join).



of two sets is equal to set of all elements that are both in A and B ( think inner join).


Disjoint or mutually exclusive

Two sets that have no elements in common


General addition rule for two sets

a. If not disjoint
i. | S u T | = |S| + |T| - |S ^ T|
b. If disjoint
i. |S u T| = |S| + |T|
ii. Since |S ^ T| = 0


Principle of Multiplication

If a first object may be chosen in m ways and a second object may be chosen in n ways, then there are mn ways of choosing both objects.
i. 5 entrees 3 desserts = 15 different meals
ii. 8 consecutive coin flips = 2^8 possible outcomes



Order matters!

If a set of n objects is to be ordered from 1st to nth, there are n choices for the 1st object, n-1 choices for the 2nd object, n-2 choices for the 3rd object etc…. until there is only 1 choice for the nth object.



Order does not matter!

n! / k! (n – k)!


Define discrete probability.

Experiments that have a finite number of outcomes. An event is a particulate set of outcomes.


Probability that an event occurs P(E)

a. P(E) = 0 ; then E is impossible
b. P(E) = 1 ; then E is the set of all possible outcomes
c. If F is a subset of P(E) then P(F) < P(E)
d. For experiments in which all of the individual outcomes are equally likely, the probability of an event E is = # outcomes in E / total number of possible outcomes.


Probabilities for combined events

Given an experiment with events E and F
i. P(not E) = 1 – P(E)
ii. P(E or F) = P(E) + P(F) – P(E and F)
iii. P(E and F) = P(E)*P(F) if not mutually exclusive


Independent Event

Two events A and B are said to be independent if the occurrence of either event does not alter the probability the other event occurs.


P(A or B)

P(A) + P(B) – P(A)*P(B)


P(A and B)



Special Addition rule for mutually exclusive events P(A or B)

P(A and B) = 0

Then P(A) + P(B)