GRE Math Formulas

Distributive property

a+c/b = a/b + c/b

Multiplying fractions

a/b x c/d = ad/bc

Dividing fractions

a/b / c/d = a/b x d/c

• LCF

Squares of fractions

(a/b)² = a²/b²

Square roots of fractions

√x/y = √x / √y

If 0 < x < 1, it must be true that…

x² < x < √x

General form of a quadratic equation

ax² + bx + c = 0

Factoring a quadratic

x² + bx + c = 0 factors to (x+p)(x+q)

• p & q must MULTIPLY to c

• p & q must ADD to b

Quadratic identity: (x+y)²

(x+y)(x+y) = x² + 2xy + y²

Quadratic identity: (x-y)²

(x-y)(x-y) = x²- 2xy + y²

Quadratic identity: (x+y)(x-y)

x² - y²

Difference of squares

a² - b² = (a+b)(a-b)

• Ex: 9x²-16 = (3x+4)(3x-4)

Even/odd rules for addition/subtraction

• odd + odd = even

• even + even = even

• even + odd = odd

• odd - odd = even

• even - even = even

Even/odd rules for multiplication

• even x even = even

• even x odd = even

• odd x even = even

• odd x odd = odd

Even/odd rules for division

• even/odd = even

• odd/odd = odd

• even/even = even or odd

Prime numbers up to 100

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

Factors

If y divides evenly into x, we say y is a factor of x

Multiple

The product of that number an any integer

A number is divisible by 2 if…

the ONES digit is 0, 2, 4, 6, or 8

A number is divisible by 3 if…

the sum of all digits is divisible by 3

A number is divisible by 4 if…

the last TWO digits of a number are a number divisible by 4

A number is divisible by 5 if…

the last digit is a 0 or a 5

A number is divisible by 6 if…

the number is divisible by both 2 and 3

A number is divisible by 8 if…

the last THREE digits are a number divisible by 8

A number is divisible by 9 if…

the sum of all digits is divisible by 9

A number is divisible by 11 if…

the sum of the odd-numbered digits MINUS the sum of the even-numbered digits is divisible by 11

Prime factorization

Finding which prime numbers multiply together to make the original number

One method:

1. Divide # by smallest prime possible; note this

2. Divide result by smallest prime possible (or, factor the result)

3. Keep going until all factors are prime numbers

• Ex: 147 → 147/3 = 49; 49/7 = 7

  • Prime factorization: 3 × 7²

Finding the total number of factors in a particular number

1. Find the prime factorization of the number

2. Add ONE to the value of each exponent

3. Multiply the results and the product will be the total number of factors

• Ex: Number of positive factors of 240

  • 240 = 2⁴ × 3¹ × 5¹

  • (4+1)(1+1)(1+1) = 20

  • 240 has 20 positive factors

Finding the LCM

1. Find the prime factorization of each integer

2. Of any repeated prime factors, take the one with the LARGEST exponent

3. Multiply together what results from Step 2

• Ex: LCM of 24 & 60

  • 24 = 2³ × 3¹

  • 60 = 2² × 3¹ × 5¹

  • 4 × 3 = 12

Finding the GCF

1. Find the prime factorization of each number

2. Of any repeated prime factors, take the one with the SMALLEST exponent

3. Of what’s left, take all non-repeated prime factors of the integers

4. Multiply together what results from Steps 2 & 3

• Ex: GCF of 24 & 60

  • 24 = 2³ × 3¹

  • 60 = 2² × 3¹ × 5¹

LCM x GCF

If the LCM of x and y is p, and the GCF of x and y is q, then xy = pq

Any factorial greater than or equal to 5!…

will always have 0 as its unit digit

Trailing zeroes

The number of (5×2) pairs in the prime factorization of that number

• Ex 1: 520 → 52 × 10 = 52(5×2), so one trailing zero

• Ex 2: 5,200 → 52 × 100 = 52(5 × 2)(5 × 2), so two trailing zeroes

Patterns in unit digits: power of 0

All powers of 0 end in 0

Patterns in unit digits: power of 2

The unit digits will follow the pattern 2-4-6-8

Patterns in unit digits: power of 3

The unit digits will follow the pattern 3-9-7-1

Patterns in unit digits: power of 4

The unit digits will follow the pattern 4-6-4-6

Patterns in unit digits: power of 5

All numbers end in 5

Patterns in unit digits: power of 6

All numbers end in 6

Patterns in unit digits: power of 7

The unit digits will follow the pattern 7-9-3-1

Patterns in unit digits: power of 8

The unit digits will follow the pattern 8-4-2-6

Patterns in unit digits: power of 9

The unit digits will follow the pattern 9-1-9-1

Perfect square

A number such that all of its prime factors have EVEN exponents

Perfect cube

A number such that all of its prime factors have exponents that are divisible by 3

Two consecutive integers will never…

share any prime factors; the GCF is 1

Multiplying radicals

^m√a x ^m√b = ^m√ab

AND

^m√ab = ^m√a x ^m√b

Dividing radicals

^m√a / ^m√b = ^m√a/b

AND

^m√a/b = ^m√a / ^m√b

Adding and subtraction radicals

√a+b DOES NOT = √a + √b

• Solve inside first, then square root

Taking the square root of a number and/or binomial

• √x² = |x|

• √(x+y)² = |x+y|

Multiplication of like bases

(x²)(x^b) = x^a+b

Division of like bases

(x^a) / (x^b) = x^a-b

Power to a power rule

(x^a)^b = x^ab

Multiplication of different bases and like exponents

(x^a)(y^a) = xy^a

Division of different bases and like exponents

x^a / y^a = (x/y)^a

Radicals as exponents

√x = x_^1/2

Multiple square roots

^m√ⁿ√x = (x^1/n)^1/m = x^{1/n x 1/m} = x^1/nm

Raising a base to a negative exponent & vice versa

x^-y = 1/x^y

Addition and subtraction of like radicals

• Add outside #s together

• Convert radical to exponent and sum to a base with exponent

• Multiply

Equations with one absolute value

Solve equation twice: (1) for the case in which the expression inside the bars is positive, and (2) for the case in which the expression inside the bars is negative

Adding absolute values

ALWAYS TRUE: |a+b| is less than or equal to |a|+|b|

When |a+b| = |a|+|b|…

1. One or both quantities are 0

OR

2. Both quantities have the same sign

Subtracting absolute values

ALWAYS TRUE: |a-b| is greater than or equal to |a|-|b|

When |a-b| = |a|-|b|…

1. The second quantity is 0

OR

2. Both quantities have the same sign and the absolute value of |a-b| is greater than or equal to the absolute value of |a|-|b|

Profit

Profit = total revenue - total cost

OR

Profit = total revenue - (total fixed costs + total variable costs)

Simple interest

Principal x Rate x Time

Compound interest

A = P(1+r/n)^nt

• A = future value

• P = principal

• r = rate of interest

• n = number of times per year interest is compounded

• t = time in years

Distance

Rate x Time

Work

Rate x Time

“Percent less than”

Final = (1 - % less than/100) x (Initial)

“Percent greater than”

Final = (1 + % greater than/100) x (Initial)

“Percent change”

((Final Value - Initial Value)/Initial Value) x 100

Counting the number of integers in a set of consecutive integers (inclusive)

(Highest number - lowest number) + 1

Counting the number of multiples in a set of consecutive integers (inclusive)

((Highest multiple - lowest multiple)/Given number) + 1

Weighted average/mean

• Where dp = data point

(dp 1) x (frequency of dp 1) +…+ (dp n) x (frequency of dp n) / total frequency of dp’s

When adding/subtracting a constant to each tern in a set of numbers, the standard deviation…

does not change

When multiplying/dividing a constant to each tern in a set of numbers, the standard deviation…

will also be multiplied/divided by that amount

Acute angle

Less than 90°

Right angle

90°

Obtuse angle

Greater than 90° but less than 180°

Straight angle (forms a line)

180°

Supplementary angles

Both angle measures add up to 180

An exterior angle of a triangle is equal to…

the sum of its two remote interior angles

Parallel lines intersected by another line (transversal)

• Vertical angles are equal (angles diagonal from e/o)

• Corresponding angles are equal (angles in the same “spot” above and below)

Area of a triangle

A = (base x height)/2

OR

A = ½(bh)

Triangle inequality theorem

In any triangle, the sum of the lengths of any two sides is GREATER THAN the length of the third side

Pythagorean theorem

a²+b²=c²

Pythagorean triples/triplets

• 3-4-5

• 5-12-13

Isosceles right triangle

Two equal angles, two equal sides, and third angle is 90°

45-45-90 triangle

Ratios:

• X for each leg

• X√2 for hypotenuse

30-60-90 triangle

Ratios:

• X for vertical side (opposite of 30)

• X√3 for horizontal side (opposite of 60)

• 2x for hypotenuse

Area of an equilateral traingle

½ of side x ½ of side x √3

When are triangles congruent?

• SSS

• SAS

• ASA

• AAS

When are triangles NOT congruent?

• AAA

• ASS

Similar triangles

Two triangles that share the same three angles; the proportions of the sides are the same

Cutting an equilateral triangle in half produces…

two identical 30-60-90 triangles

Parallelogram

• Opposite sides are equal

• Opposite angles are equal

• A = bh

  • Height must be found by creating a triangle if not already provided

• A rhombus is a regular version of this shape

Longest line segment of a rectangle

The diagonal

• Diagonal = √L²+W²

The diagonals in a square are…

perpendicular

How to find the maximum area of a given shape

Make it regular (all sides equal)

• A rectangle with the maximum area is a square

Trapezoid

• Top and bottom are parallel

• A = (base₁+base₂) x height / 2