Number Properties GMAT

NUMBER PROPERTIES-NUMBER PROPERTIES-NUMBER PROPERTIES-NUMBER PROPERTIES-NUMBER PROPERTIES-

1.  Divisibility & Prime Factors

The Sum and the Difference of 2 numbers are also Divisible.

Prime numbers...

2,3,5,7,11,13,17,19,23,29,31,37,41,47,53,59,61,67

Factor Foundation Rule...

Numbers are built on building blocks and the factors of those blocks are also factors of the main number.

Prime Box..

Factors of a number can be created by multiplying different combinations of the prime factors together.

Greatest Common Factor and Least Common Multiple..

GCF-the largest number by which two integers can be divided.  Find the product of lowest COMMON prime factors. 

LCM-the smallest number that is a multiple of two integers.  Find the product of ALL prime factors of both numbers using the higher power of the repeated factor.

2.  Odds & Evens Strategy

Addition & Subtraction Rules...

Add 2 odds or add 2 evens and an EVEN you shall see,

but add an odd with an even and oh how ODD t'will be.

Multiplication Rules...

Just 1 EVEN number in a multiplication set

and no matter all the others, an EVEN you will get. 

The Sum of 2 Primes....

All prime numbers are odd expect the number 2.

So the sum of any 2 primes will be even.

If the sum of two primes is odd, one of those primes must be the number 2. 

3.  Positives & Negatives Strategy

Absolute value...

|-5| = 5

|0| = 0

Absolutely Positive!!!

A double negative = A positive....

two negatives yield a positive.

Multiplying & Dividing Signed Numbers....

If the Signs are the Same, answer's poSitive

but if Not, the answer is Negative.

4.  Consecutive Integers Strategy

Be inclusive and Add 1 before your done...

Counting consecutive integers.

How many integers are there from 14 to 765, inclusive?

765-14 plus 1 = 752

Because 14 has been subtracted away. 

The Sum of Consecutive Integers...

Q. What is the sum of all the integers from 20 to 100, inclusive?

3 steps.

1) average the first and last number.  100+20= 120/2=60

2) count the number of consecutive integers (as done above) 100-20 plus 1 = 81

3)Multiply the average number and the number of terms to find the sum.  60x81 = 4,860

The average term in a consecutive set....

What is the average of all the integers from 20 to 100, inclusive? 

100 + 20 = 120 /2 = 60

or

Sum of all integers  = 4860   =  60  

# of integers             81

*the average of an odd set of consecutive integers is always an integer.  But the average of an Even consecutive set of integers is never an integer.  Because there is no true middle number.

Special Products....

In a series of X consecutive integers the product is always divisible by X.

Special Sums...

Only applies to ODD amount of consecutive integers.

In a series of 3 consecutive integers the sum is always divisible by 3. 

Consecutive integers and divisibility...

If there is one EVEN integer in a consecutive series, the product of the series is divisible by 2.

If there is two EVEN integer in a consecutive series, the product of the series is divisible by 4.  

5. Exponents Strategy

Base of 0 or 1...
An exponential expression w/ a base of 0 always yields 0, regardless of the exponent.

An exponential expression w/ a base of 1 always yields 1, regardless of the exponent.

A Fractional Base...
When the base of an exponential expression is between 0 & 1 an interesting thing occurs.
As the exponent increases the value of the expression decreases.

The Sign of the Exponent...
If the exponent is negative simply do the inverse.
5

GMAT math hints and basics

Probability

More than 1 event "and" vs "or"



AND

X & Y are both independent, whats the prob they will both occur?

MULTIPLY the two probs together.



OR

Whats the prob a die will land on 4 OR 5?

ADD the probs together. 1/6 +1/6 = 1/3



NOT MUTUALLY EXCLUSIVE events (both events can occur)

coin and die are flipped and rolled once. Whats the prob of a 3 or landing on heads?

ADD the prob of either event occuring and SUBTRACT the prob of BOTH events occurring.



STATISTICS


Evenly spaced sets-

To find the average or middle number of an evenly spaced set (add the first and the last number together and divide by two).



Roots in the denominator:

Multiply the denominator by its complement in both the nom/denom. ie 2/5+2sq6 mult n/d by 5-2sq6.

Then the d is 25-4(6)=1



WORD TRANSLATIONS

-Algebra equations

Organize Variables into charts



-Rates and Work

matching chart units

clicking charts (rtd chart) use the person with the most x+1

The averafe rate fallacy

working together:Add the rates

Population chart



-Ratios

The unknown multiplier

Proportiojns



-Combinatorics

Anagram Grid



-Probablity

AND vs OR

The domino effect

1-x shortcut



-average

The average law



-overlapping sets

double set matrix (comparing two things)

Venn Diagram:Inside-Out



VIC's

You should know that

y + x = x + y and y – x = -(x – y), it can be stated that

(y + x) (y – x)

= -

(x + y) (x – y)

Venn Diagram
When to use it.
When you know the amount of all three use either 0 or the number.
When you don't know the number of all three use the -X to determine the otehrs.

Multiplying Fractions
1/3 x (1/4)n
put N in the numerator and 12 in the denominator.

Ratio's
If the ratio x:y = 3:4 what is the ratio if x and y are increased by 5 each?
A. You cannot do it. Because you don't know what x and y are.
in digits questions set one variable as

Exponents and roots
All even exponents can be treated as squares. X^22 can be written (x^11)(x^11).
also X^22 - Y^48 can be wriitten (X^11 + Y^24)(X^11 - Y^24)

Radicals in the denominator.
multiply top and bottoom by the complement.

Two fractions one with a variable in the denominator and one with a number. They are set equal to each other. To simply cross multiply BOTH the denominators!!!

number props
every integer is a factor of itself.
r and s are positive integers. is r/s an integer?
if every factor of s is also a factor of r, then it is.
if every prime of s is a prime of s then its maybe.

canceling out
72832x/3232x = 72832/3232 The x's cancel out!!!!!!!!

decimal to fraction.
convert .0625 into a fraction.
.0625/1
make .0625 a whole number. Then move the decimal point on the 1 the same number of places.
625/10,000

degree of each side of a right triangle
45,45,90

exponents
1^x = 1 always regardless of x
0^x = 0 always regardless of x

opposite angles that intersect have the same degree angle.

Volume of a Cube
s x s x s = S^3 or the volume of a cube. All sides are equal.

sequence
3n+4 should be > than 50. whats n?
n is the position of the thing in the sequence. so n=1 is the furst number etc.

I. Number Properties
a. Consecutuve integers strategy
1. sum of consecutive integers (inclusive). whats the sum of all integers from 20 to 100 inclusive?
average the first and last term to get the middle number. 20+100=120/2=60
count the number of terms 100-20=80+1=81
multiply the middle number or the average by the number of terms 60x81=4860

2. what is the average of all integers from 20 to 100 inclusive?
sum of all integers/number of terms=average Same as above.


3. how to find the middle number in a set of integers from 400 to 600?
add the first and last terms togetehr and divide by 2.

*average of an odd number of consecutive integers will always be an integer. avg of an even #of integers will never be an integer.

4. special products
every number is divisible by the factors of its factors.
if there is always a multiple of 3 in a set of consecutive integers then the product of the integers will always be divisible by 3.
*the product of a set of 3 consecutive integers will always be divisible by the number of consecutive terms.
the product of 4 consecutive integers is divisible by 4, etc.

5. Special Sums rule (only works for odd # of consecutive integers).
the sum of a set of consecutive integers with an odd amount of terms will always be a multiple of the number of terms.
ie. 1+2+3+4+5=15 and five numbers so, 15 is a multiple of five

*the 2 above rules apply to ALL consecutive sets of integers except special sums is only odds. But 2x+2 + 2x+3 +2x+4 still qualifies for the special sum rule and the sum of 6x+9 will be a multiple of 3.

6. consecutive integers and divisibility by 2's.
if their is 1 even integer in a consecutive series the product is divisible by 2. If their is 2 even integers in a consecutive series the product is divisible by 4,. etc etc.

prime numbers
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41

if she touches those kids or pushes them away one more time im gonna have to say something. for the kids benefit.

geometry
A hypotenuse is the longest side of a right triangle, the side opposite the right angle.

where x and y are each a side of the triangle.

Exponents
Successive exponents have many ways to write them.
(3^2)^4=
(3x3)(3x3)(3x3)(3x3) = (3^2)(3^2)(3^2)(3^2) = 3^8 = 3^2x4

arithmetic
if x is m percent of y, then in terms of m, y is what percent of x?
this says in english
x = (m/100)y

gmat verbal "sentence correction" notes

GMAT Sentence Correction section

The 3 C's strategy

1) Correctness-
2) Concision-fewer words is better
3) Clarity-did u understand after reading it once?

Concision

Use of a phrase is discouraged when a single word will do.
(have differences------>differ)

Avoid Redundancy
If a word can be removed without changing the meaning of the sentence then remove it.

Clarity

If a sentences meaning is difficult to understand it is probably wrapped in the answer choices.
The answer should not change the original meanig of the sendtence.
Instances of altered meaning:
1) Word placement
the position of a word can alter a sentences meaning. 
2) known vs unknown
3) multiple meanings
4) "such as" vs "like"

Stopped in 3 C's section on page 17


Subject-Verb Agreement
The KEY is to (first find if the subject of a sentence is singular or plural then you can determine the verb)

A singular subject requires a singular verb form
A plural subject requires a plural verb form

Elminate the middleman.
junk that tries to disguise the suject from the verb is usually in the middle.

"of" middleman
subject of xx xxx xxx verb.  everything after the "of" is junk until the verb.

"And" vs Additive
"and" can be used inbetween 2 singular subjects to make them plural hence a plural verb.
An "additive" phrase such as: in addition to, along with, together with, including.
The "additive" phrase keeps two or more signular subjects singular hence singular verbs.
ONLY THE WORD "AND" CAN CHANGE A SINGULAR SUBJECT INTO A PLURAL ONE.

"Or", "Either...Or" & "Neither...Nor"
These sentences contain TWO subjects.  One of which can be singular and the other plural.
Use the subject closest to the verb to determine if the verb should be signular or plural.
"either" & "neither" in sentences by themselves are considered signular.

Collective nouns are singular.
Collective nouns refer to a group of pleaople and do not end with an "S".
administration, army, audience, class, crowd, facualty, team.

Indefinite Pronouns: usually singular
pronouns are words that replace other nouns or pronouns.
an indefinite pronoun is not specific about the thing to which it refers.
"Anyone" is an example.
The following are "singular indefinite pronouns:
Anyone, Anybody, Anything
Everyone