TTP Essential GMAT Quant Skills

Finding the LCM

(1) Find the prime factorization of each integer. That is, prime factorize each integer and put the prime factors of each integer in exponent form. (2) Of any REPEATED prime factors, take only those with the LARGEST exponent. For example, given a 3^2 in one # and a 3^3 in another, take the 3^3. If left with the same of two numbers, take one of the duplication. (3) Of what is left, take all NON-REPEATED (4) Multiply together what is found in 2 and 3 LCM of 24 and 60 (1) 24 = 2^3 x 3^1 and 60 = = 2^2 x 3 x 5 (2) 2^3 x 3 (3) 5 (4) 2^3 x 3 x 5 = 120

Finding the GCF

(1) Find the prime factorization of each number. (2) Of any REPEATED prime factors, take only those with the SMALLEST exponent. (Note: if no repeated factors found, then GCF = 1) (3) Multiply together the numbers that you found in step 2; this product is the GCF. GCF 24 and 60 (1) 24 = 2^3 x 3^1 and 60 = = 2^2 x 3 x 5 (2) 2^2 and 3^1 (3) 2^2 x 3^1 = 12

Finding count/number of factors in a number

(1) Find the prime factorization of the number (2) Add 1 to the value of each exponent. Then multiply these results and the product will be the total number of factors for that number. Note: for odd take only the odd bases exponents, and for even take total factors minus odd factors.

LCD approach to comparing fractions

(1) If given a set of different fractions, multiply top and bottom of each to get all with the same common denominator (2) The larger the numerator, the larger the fraction after all fractions have the same denominator Note: A similar approach works to getting all fractions with the same numerator, but instead for step two the smaller denominator the larger the fraction

Multiplication of decimals

(1) Ignore the decimals points and treat the #s as whole numbers (2) Count the total decimal points for EACH number (ex: 0.02 = 2) (3) In the product of 1, move the decimal places to the sum of numbers counted in (2) Ex: .02 x .02 (1) 2x2 = 4 (2) 4 decimal places because .02 is 2 and .02 is 2 (i.e. the 0 and 2 after the decimal place (3) .0004 because counted 4 in step 2, so need 4 total places after the decimal point NOTE!!!!: A shortcut to answer questions can be to simply count the decimal places instead of having to actually perform the multiplication. In counting places in 2, the answer MUST have that amount of decimal places, so if there is only one answer with that then that is the answer.

Squares of Fractions

(a/b)^2 = a^2/b^2

Common quadratic identities

(x+y)^2 = x^2 + 2xy + y^2 [Square of a sum] (x-y)^2 = x^2 - 2xy + y^2 [Square of a difference] (x+y)(x-y) = x^2 - y^2 [Difference of two squares]

Any factorial >5! has what as it's last number/units digit

0

0!

1

Reciprocal of ANY number

1 divided by ANY number is the reciprocal of that number In general 1/a/b = b/a

Representing a three digit integer in algebraic form

100X + 10Y + Z Ex: 312 = 100*3 + 10*1 + 2 This pattern continues

Prime numbers less than 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

Number 2 units digit powers pattern

2-4-8-6 ex: 2^1 = 1, 2^2 = 4, 2^3 = 8, 2^4 = 16

Number 3 units digit powers pattern

3-9-7-1

Number 4 units digit powers pattern

4-6 pattern All positive odd powers of 4 end in 4, and all even powers of 4 end in 6.

Number 7 units digit powers pattern

7-9-3-1

Number 8 units digit powers pattern

8-4-2-6

Number 9 units digit powers pattern

9-1 pattern All positive odds end in 9, and all positive evens end in 1

Perfect cubes

A perfect cube, other than 1 and 0, is a number such that all of it's prime factors have exponents that are DIVISIBLE by 3 27 = 3^3

Perfect squares

A perfect square, OTHER THAN 1 AND 0, is a number such that all of it's prime factors have even exponents. Ex: 144 = 2^4 x 3^2

Factoring Quadratics

A quadratic equation in the form ax^2 + bx + c can be rewritten as a product of two factors called the "factored form". This form resembles (x + ?)(x + ?) (What adds to bx, and multiplies to c) Ex: X^2 - 3x - 28 = (x-7)(x+4)

Adding or subtracting a constant to the numerator and denominator

Adding the same number to top and bottom moves the # closer to 1 Subtracting the same number to top and bottom moves the number AWAY from 1 (note: for subtraction, this doesn't hold true when the numerator becomes <0 (i.e. negative))

Number 5 units digit powers pattern

All positive integer powers of 5 end in 5

Number 6 units digit powers pattern

All positive integer powers of 6 end in 6

What does percent mean?

DIVIDE THAT SHIT BY 100. Whatever it is, it quite literally mean to divide by 100. For example, x percent of y is what? x/100 TIMES y

Determining if a decimal is terminating

Decimal equivalent of a fraction is terminating if and only if the denominator of the REDUCED fraction has a prime factorization of only 2s or 5s, or both.

Comparing decimal sizes methodically

Ex: .9098 vs .907 (1) .9 = .9 (2) .00 = .00 (3) .009 > .007 So the answer is (1) is larger because thousandths place is larger in the first number than second, and all places before it are equal.

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

Factors of 16 are 1, 2, 4, 8, and 16

Divisibility rule for 6 and 12

For 6, determine if divisible by 2 and 3 For 12, determine if divisible by 3 and 4

LCM x GCF

If the LCM of x and y is P, and the GCF of x and y is Q, then pq=xy Example for 24 and 60 24 x 60 = 12 x 120

Adding/subtracting decimals

Line up the decimals along the decimal point, and add/subtract as normal

Range of possible remainders

Must be a non-negative integer that is less than the divisor

Converting decimals to percents

Note: perform the inverse for converting a percent to a decimal

Divisibility rules for 3 and 9

Sum of the numbers divisible by 3/9

PEMDAS (hint: left to right for certain parts of pemdas)

The GMAT WILL try to trick you on the multiplication/division left to right part of pemdas.

Divisibility rule for 4

The last 2 digits are divisible by 4

Trailing zeroes

The number of trailing zeroes of a number is equal to the number of (5 x 2) pairs in the prime factorization of that number. Ex: 100 = 10x10 = [5x2]x[5x2] = two trailing zeroes

Dividing Fractions

To divide fractions, invert the second one and multiply

Two consecutive integers

Two consecutive integers will NEVER share ANY prime factors. Thus, the GCF of two consecutive integers is 1.

Distributive Property

[a+c]/b = a/b + c/b

Adding different denominators

a/b + c/d = [ad + bc]/bd

Comparing fractions bow tie method

a/b > c/d IF ad > bc Ex: 3/4 > 5/7 beacause 21>20

Multiplication rules for even and odd numbers

even x even = even even x odd = even odd x odd = odd

Zero product property

if a x b = 0, then a = 0 or b = 0

Divisibility rule for 8

if the last three digits are divisible by 8

Even/odd rules for addition/subtraction

odd + odd = even even + even = even even + odd = odd odd - odd = even even - even = even

Square roots of fractions

sqrt(x/y) = sqrt(x)/sqrt(y)

Expression set equal to 0

x CAN equal 0 ex: x(x+100) = 0 x = 0, or x+100 = 0

Division Formula

x/y = quotient + remainder / y

If 0<x<1 then what's the order for x^2, x and sqrt(x)

x^2<x<sqrt(x)