GMAT - Quant Chapters 1-5

properties of 1

1) One is a factor of all numbers, and all numbers are multiples of 1. 2) 1^any # = 1 3) Multiplying or dividing by 1 does not change the value 4) Odd # 5) Only # with exactly 1 factor. 6) not a prime number

Associative Property of Addition

(a+b)+c=a+(b+c)

Common quadratic identities

(x+y)^2=(x+y)(x+y)=x^2+y^2+2xy (x-y)^2=(x-y)(x-y)=x^2+y^2-2xy (x+y)(x-y)=x^2-y^2

If x cannot equal y, then (x-y)/(y-x)=?

-1

square root

-consider only the non-negative number -Square root of a positive will be positive -Square root of a variable will produce = |x| or +/- x

0!=? 1!=?

0!=1 1!=1

properties of 0

1) only # that is neither +/- 2) a multiple of every integer 3) is an even # 4) a/0 = undefined 5) 0/a = 0 6) 0^2=0. Square root is also 0. 7) Only number that is equal its opposite. 0=-0 8) not a factor of any # except itself

"1 over a fraction"

1/a/b= b/a

Examples of Perfect Squares

0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,225

A Prime number only has 2 factors: 1 and itself. Prime numbers through 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

What will perfect squares never end in?

2,3,7,8. Always end in 0,1,4,5,6,9

Patterns in Units Digits

2: 2-4-8-6 3: 3-9-7-1 4: 4-6 5: 5 6: 6 7: 7-9-3-1 8: 8-4-2-6 9: 9-1 (all even powers of 9 will end in 1 and all odd powers will end in 9) *all integers > 9 follow the same digit pattern as above

Prime Factors vs unique Prime factors.

2^1 x 3^2 x 5^1 Prime factors = 4 Unique Prime factors = 3 (Number of unique prime factors doesn't change no matter how many factors the number is raised to.)

Don't assume two equations are sufficient to determine the values of two variables, especially when one equation is just a multiple of the other equation.

6x+3y=15 2x+y=5. Cannot use these together to solve for a specefic variable.

Rules for even/odd numbers

Addition: Even + even Odd + Odd = Even Even - even odd - odd = Even Even + odd or Even - odd = odd Multiplication: Even x odd/even=even Odd x Odd=Even Even x Odd x Odd = Even Odd x Odd x Odd= Odd Division: Even/odd=even. Odd/odd=odd. Even/even=odd or even

Definition of: Integer, Whole number, Constant

An integer is a number (positive or negative) without a decimal or fraction. No remainder. A whole number: are nonnegative numbers (positive numbers and 0). Constant: a number that does not change

Determining the number of trailing zeros in a number.

Count the amount of (5 x 2) pairs. The least amount of 5s or 2s will be the limiting factor. Any factorial >/= 5! will always have a 0 as the units digit.

Equation Trap: there is a quadratic equation present

Don't assume in every case that one quadratic equation is sufficient, especially if it has two unique roots. Two answers is not sufficient.

When solving equations that contain fractions, consider multiplying the entire equation by the LCM of the denominators to remove the fractions.

EX: 1/5 + n= 1/15 + 3n=--> 3 +15n=1+45n

The product of n consecutive even integers is always divisible by 2^n x n!

Ex: 2^2 x 2! = 4 x 2 = 8. For example, 10 x 12 = 120. 120 is divisible by 8.

The product of any n consecutive integers is always divisible by n!

Ex: 3 consecutive integers is always divisible by 3!. 4 is always divisible by 4!

when a perfect square ends with an even number of zeros, the square root is half the number of zeros. (Works with decimals too)

Ex: sqrt 81,000,000 -> 9000 sqrt 0.0004 -> 4/10,000 ->0.02

|x+y|=|m+n|

For absolute values that are equal they must be the same or opposites. So x+y=m+n or x+y=-m-n. Each works.

If I factor any number above 5!, what is the limiting number?

IT WILL NEVER BE 2 OR 3

Determining Leading Zeros

If 1/x and x is not a perfect power of 10 it will always be (k-1) where k = # of digits in denominator. Example: 1/800= 0.00125 --> (3-1)= 2 leading 0's If x is a perfect power of 10. Then it is k-2

LCM vs GCF

LCM = smallest positive multiple of all integers in the set. EX: 5, 2 = LCM 10 (multiply the greatest prime factor from each together.) EX: 3^2 or 3^3 take 3^3. -Provides us with the all unique prime factors of the set. -Will tell us how to solve repeating pattern questions i.e. L flashes every 32 seconds and M flashes every 12 seconds. They flashed at 8pm. When will they flash again? 96 seconds later. GCF = of the set of positive integers is the largest number that will divide into all of the numbers in the set. EX: 8,12,16 = 4 GCF

Number of Positive divisors of a number

Prime factorize the number. Take each exponent and add 1. Then multiply the numbers. Ex: 1260=2^2 x 3^2 x 5^1 x 7^1 = (2+1)(2+1)(1+1)(1+1)= 36 positive divisors in 1260.

Cube Root approximations

Root 2 = 1.3 Root 3 = 1.4 Root 4 = 1.6 Root 5 = 1.7 Root 6 = 1.8 Root 7 = 1.9 Root 9 = 2.1

square root approximations

Root 2 = 1.4 Root 3 = 1.7 Root 5 = 2.2 Root 6 = 2.4 Root 7 = 2.7 Root 8 = 2.8 Root 11 = 3.3

What do terminating decimals end in?

The denominator contains only 2's or 5's or some combination of both.

Two Consecutive Integers

They will never share the same prime factors. Thus, the GCF of two consecutive integers is 1. That is GCF (n, n+1) = 1

If b cannot = 0 then (a/b)^2 =? If x is greater than or equal to 0 and y>0, then square root of (x/y)=?

a^2/b^2 Square root of x and square root of y

common numerator

numbers that have the same numerator, such as 3/5 and 3/4. When comparing, the smaller the denominator, the larger the fraction.

You can divide by a variable ONLY when you know that the expression is not equal to zero.

x(x+1)=4(x+1) I CANNOT divide by x+1 because I don't know if this is set to 0. Correct way to solve is: x(x+1)-4(x+1)=0 (x+1)[x-4]=0 x=-1,4. Two answers.

Formula for division

x/y = Q + r/y

If I see x(x+100)=0, what could x be?

x=0 or x+100=0. Don't think that I can just divide x out of the equation. Unless stated, the answer can always be 0.

x/y = y/x is equivalent to what?

x^2 - y^2 or (x+y)(x-y)

If 0 < x < 1

x^2 < x < sqrt(x)

You can divide by a variable ONLY when you know that the variable is not equal to zero

x^2=100x. I CANNOT divide by x because I don't know if x=0. Correct way to solve is: x^2=100x x^2-100x=0 x(x-100)=0 x=0,100. Two answers.

(x^a)(x^b) (x^a)(y^a)

x^a+b (xy)^a

If we know the LCM and GCF, we know the product of x and y.

xy = LCM (x, y) x GCF (x,y)

Examples of perfect Cubes

· 0,1,8,27,64,125,216,343,512,729,1000

Divisibility Rules

· 0: No number is divisible by 0 · 1: All numbers are divisible by 1 · 2: All even numbers are divisible by 2 · 3: If the sum of all numbers is divisible by 3 then the number is as well. Ex: 135 (1+3+5=9) · 4: If the last two numbers are divisible by 4. Ex 244 are 44 which 4 divides into. 00 is divisible by four Ex 100. · 5: Numbers that end in 0 and 5. · 6: An even number whose digits sum to a multiple of 3. Ex (18=1+8=9) · 7: Take the last digit of the number, double it, then subtract the result from the rest of the number. If the resulting number is evenly divisible by 7, so is the original number · 8. If the last three digits can be divided by 8, then the number can be. EX: 1,160. à . Remember that 000 is divisible by 8. · 9. A number is divisible by 9 if the sum of digits is divisible by 9. Ex: 479,655. (4+7+9+6+5+5=36) · 10. If the ones digit is a 0, then 10 will divide into it. · 11. Take the odd numbered digits minus the even number digits. Ex: 253. (2+3-5=0) 0 works for 11. EX: 2,915 (9+5)-(2+1)=11. · 12. If a number is divisible by 3 and 4, it is divisible by 12.