GRE MATH
To find the LCM (EX: 24 & 32)
1. Find GCF. 2. Write each number as a product of the GCF x another factor ( Ex: 8x3; 8x4 3. The LCM is the product of these 3 factors. 8x3x4 =96
.50
1/2
.333
1/3
11^2
121
12^2
144
13^2
169
14^2
196
What is the only even prime number?
2
If the ratio of boys to girls is 3:5, boys are what fraction of the whole?
3/8 Total parts =8
18^2
324
Squaring rule for 5's
35^2 is between 30^2 and 40^2, so instead of 3x3 or 4x4, we use 3x4 "added" to 25, for 1225
19^2
361
The first 5 multiples of 4
4, 8, 12, 16, 20
.8333
5/6
.714
5/7
.888
8/9
Always cancel before you multiply
Always make the numbers smaller before you make them bigger.
Order of operations
GEMDAS: Grouping symbols, Exponents, Multiplication, Division, Addition, Subtraction
Divisibility rule for 9
If the sum of the digits is divisible evenly by by 9
Is 1 a prime number?
No, it does not have 2 distinct factors.
REMEMBER:
One is NOT a prime number!
Absolute value
The distance a number is from zero on a number line--always positive (except zero)
If we have addition or subtraction in both the denominator and the numerator, we can split up the numerator, but the denominator has to stay unchanged
a + b/c + d = a/c+d + b/c+d
result of subtraction
difference
Evens and odds only pertain to
integers
even + odd or even - odd
odd
Make multiplication easier
separate out numbers. Ex: 16 x 35 = 8x2x35 = 8x70=560
divisor
the number that divides the dividend
dividend
the number that is being divided
quotient
the result of division
real numbers
the set of all rational and irrational numbers
.571
4/7
20^2
400
integer quotient
5 goes into 17 3 times with a remainder of 2
Always, always, always
cancel before you multiply fractions
All % increase/decrease problems can be solved by
change in values (DELTA X)/initial amount
ILLEGAL cancellation in proportion. THIS IS NOT THE SAME AS CROSS MULTIPLYING OR DIVIDING. Proportions are set equal to one another.
diagonal cancellation 100% illegal and incorrect
If your annual % rate is 5 percent, and it is compounded, quarterly, then to find the correct %, simply
divide 5/4
No rules of odds or evens for
division
Add or subtract "likes" we get
even
As long as there is at least one EVEN factor in a product, the product will be
even
The exponents of the prime factors of a square all must be
even
even + even or even - even
even
even x even
even
even x odd
even
odd + odd or odd-odd
even
Don't forget that zero is an
even integer
If r is a factor of Q, r is a divisor of Q, Q is divisible by r, Q is a multiple of r, then
every prime factor of r is included in the prime factorization of Q
In order to be divisible by a number, you have to be divisible by its
factors and multiples of factors (example: if 2^3 is a factor of one, then the other must be divisible by 8)
If the test tells you a number is even or odd, therefore, you know it's an
integer
If we divide any integer by one of its divisors, that number is always an
integer
If you're told something has a remainder, once again, you know you're dealing with an
integer
If the test identifies a number as _____, ____, _____ or ____, you know it's an integer
integer, even, odd, prime
A perfect square always has an _____(odd or even) number of factors
odd
Add or subtract "unlikes" , we get
odd
The only way a product can be odd is if every single factor is
odd
odd x odd
odd
If a number is >1, its reciprocal is between
0 and 1
Find prime factors of large numbers:
1. List the prime factors as squares. 2. List the exponents in order of the prime factors (including one). 3. Add one to every number on our list from #2. 4. Multiply these numbers together. 5. This will give you the number of factors.
Legal Cancellation in Proportions. 9/4=3/16 (example)
1. We can cancel factors in numerator and denominator of same fraction. 2. We can cancel factors in two numerators on opposite sides. 3. We can cancel factors in two denominators on opposite sides.
.2
1/5
.1667
1/6
.143
1/7
.125
1/8
.111
1/9
Prime numbers less than 20
2, 3, 5, 7, 11, 13, 17, 19
.667
2/3
.4
2/5
.286
2/7
.222
2/9
15^2
225
Prime numbers between 20 and 60
23, 29, 31, 37, 41, 43, 47, 53, 59
16^2
256
17^2
289
EXAMPLE: 21600
2^5, 3^3, 5^2 (5, 3, 2) add 1 (6, 4, 3) multiply = 72. There are 72 factors.
EXAMPLE:
2^6 3^4 5^2 must be a perfect square b/c all of the exponents of the factorization are even
Circle Circumference Formula
2πr or πd
mixed numeral quotient
3 2/5
.75
3/4
.6
3/5
.429
3/7
Area of a triangle formula
A=1/2bh
Area of a rectangle formula
A=bh
Real numbers
ANY number on the number line
Compound Interest Formula
Amount= Investment(multiplier)^y (y=years)
We CANNOT separate a fraction by addition or subtraction in the denominator
Ex: a/b+c IS NOT EQUAL TO a/b + a/c a/b-c IS NOT EQUAL TO a/b-a/c
If P is a multiple of r, then any multiple of P is a multiple of r.
Example: 52 is a multiple of 13, thus 52+ 52 is a multiple of 13.
Divisibility by 4
If the last two digits form a number divisible by 4, then the number is divisible by 4. For example, 316 is divisible by 4 since 16 is divisible by 4.
To test whether a prime number less than 100 is prime, we only need to check if it is divisible by one of the prime numbers less than 10.
If the number is NOT divisible by any prime number less than 10, then the number must be prime.
Canceling is a form of division.
It is the division of the same factor in the numerator and the denominator.
LCM= #1 x #2/GCF
Make sure to cancel first!
Factors
Numbers that are multiplied together to get a product
If P and Q are multiples of r, then
P+q and P-q must also be multiples of r. Example: 700 is a multiple of 7. 49 is a multiple of 7, therefore 749 must be a multiple of 7 as is 700-49=651
In general, if a problem talks about a P % increase, the multiplier is 1 + p%.
So if you need a 30% increase, you multiply by 1.30
greatest common factor (GCF)
The largest factor that two or more numbers have in common. Same as greatest common divisor.
remainder
The amount left over when one number is divided by another.
divisor
The number by which another number is divided EVENLY--must produce an INTEGER.
Divisibility rule for 6
The number is divisible by both 2 and 3
least common multiple (LCM)
The smallest multiple (other than zero) that two or more numbers have in common.
multiplier for a decrease is 1- the percentage.
To find a 30% decrease, multiply by 1-.30=.70
What is 80% of 200? (break this down into algebraic terms)
What = x, is means equals, 80% becomes multiplier, of means .8 x 200.
Integers
Whole numbers and their opposites (. . . -3, -2, -1, 0, 1, 2, 3. . .)
a(b+c) =
ab + ac
A(b-c)
ab-ac
Integers
all + and - WHOLE numbers on a number line
change from % to fraction
put % over 100, then simplify Ex: 20% = 20/100 =1/5
result of division
quotient
17/5 quotient = _______, remainder = _______
quotient is 3. remainder is 2
multiplier =
ratio of new price/old price; then change to a % If you get 1.x, then the x is the % increase. If you get x for a % decrease, then 1-x =% you want.
To find only the ODD prime factors, repeat the above list except IGNORE
the even number's factors. In other words, discard the 2 and it's exponent of 5. List only (3, 2), add 1 (4,3) multiply, get 12. There are only 12 odd factors of 21600.
If P and Q are multiples of r
then the product P*Q must also be a multiple of r. Example: 24 & 80 are both multiples of 8. We can ADD them 104 = multiple of 8. We can SUBTRACT them 80-24=56 is a multiple of 8 or we can MULTIPLY THEM 80*24=1920 is a multiple of 8. BUT WE CANNOT DIVIDE THEM.
If you increase a numerator, but keep your denominator the same, your fraction is
bigger
If you change the denominator, but keep the numerator
bigger denominator = smaller fraction
% means
divide by 100
the result of multiplication
product
If P>q, then
s/p <s/q
Result of addition =
sum
factor and divisor are
synonyms. We usually use factor in x and divisor in division, but they mean the same thing
.25 =
1/4
.375
3/8
.8
4/5
.444
4/9
Squaring rule
40^2 =4^2 x 10^2 = 16x100=1600
n^2+ n +(n+1) = (n+1)^2
41^2 = 40^2 + 40+41 = 1600 + 81=1681
.625
5/8
.555
5/9
56 is what % of 800?
56 = x800 x=.07 which is our multiplier, so we convert it to a percent which is 7%
.857
6/7
n^2-n-(n-1)
69^2 = 70^2-70-69 = 4900-70-69=4761
.875
7/8
.777
7/9
Double / Half Rule
84 x 50 1/2 one, then double the other. 1/2 of 84=42. 50 doubled = 100 42 x 100 =4200
If a number is < 1, its reciprocal is
>1
We CAN separate a fraction into two fractions by addition or subtraction in the numerator.
Ex: a+b/c = a/c + b/c OR d-e/f = d/f-e/f
GEMDAS (Order of Operations)
Grouping symbols, exponents, multiply/divide, add/subtract
irrational numbers
Numbers that cannot be expressed as a ratio of two integers. Their decimal expansions are nonending and nonrepeating.
To divide a number by 5
double it, then divide by 10
In general, if there are n compounding periods each year, we divide the annual percentage rate by n to get the percentage for
each individual compounding period. Multiplier would be (1 + I/100n))^ny (n compounding periods in y years
As a general rule when translating words to math, the word "is" means _____ and the word "of" means_____
equals, multiply
Divisibility rule for 3
if the sum of the digits is divisible by 3
If you change both the numerator and denominator (if numerator increases and denominator decreases), the fraction
increases
multiples and factors have
inverse relationships
A decimal form of a percent is called the ____ for that %.
multiplier
To change from decimals to percent
multiply by 100 Ex. .62 x 100 = 62%