QUANT (GMAT)

Math Fundamentals: Order of Operations - PEMDAS???

The following is the order of operations for working out any algebraic expression, remembered as PEMDAS: ❶ Parentheses ❷ Exponents ❸ Multiplication (together with Division, from left to right) ❹ Division (together with Multiplication, from left to right) ❺ Addition (together with Subtraction, from left to right) ❻ Subtraction (together with Addition, from left to right).

Roots: Perfect squares???

The following table lists the first 17 perfect squares - memorize these by heart. Integer (a)Perfect square (a²). Ex: 1=1, 2=4, 3=9, 4=16, 5=25, 6=36, 7=49, 8=64, 9=81, 10=100, 11=121, 12=144, 13=169, 14=196, 15=225, 16=256, 17=289, 18=328, 19=361, 20=400, 21=441, 22=484, 23=529, 24=576, 25=625, 26=676, 27=729, 28=784, 29=841, 30=900.

Fractions - Denominator Dynamics - increase the denominator???

The greater the denominator, the smaller the fraction (under the same numerator). Ex: ½ > 1/3 > 1/4. The reverse is also true: The smaller the denominator, the greater the fraction (under the same numerator). Ex1: 1/10 < 1/5 < ½ < 1/(½) = 2/1.

Plugging In - DOZEN F???

Make sure your answer is always true by plugging in DOZEN F. These numbers, in turn, cause the answers to behave differently, exposing the sometimes answers while leaving only the always true answer. Some of these numbers overlap with the bad numbers. Remember the DOZEN F: (a) Different (e.g. even vs. odds, prime vs. multiple, fractions vs. integers, positive vs. negative, etc.). (b) One. (c) Zero. (d) Equal numbers for different variables. (e) Negatives (f) Fractions.

Integers: Adding and Subtracting Multiples???

Multiple of n ± multiple of n = MUST be a multiple of n. multiple of n ± NOT multiple of n = CANNOT be a multiple of n. NOT multiple of n ± NOT a multiple of n - May or May NOT be a multiple of n.

POE Stands for ???

Process Of Elimination

Sets: Basic terms - sample space, set, complementary set, subset???

SAMPLE SPACE - The group from which all sets are selected. COMPLEMENTARY SET - All members of the sample space which are NOT members of a designated set. SUBSET - If all members of set B are also members of set A then B is a subset of A.

Data Sufficiency: Basic Work Order???

(A) Always begin by reading the question stem and try to figure out the issue of the problem (i.e., which piece of data is required to answer the question). (B) Read statement ❶ alone and decide sufficient/insufficient. (C.) POE like so: If statement ❶ is sufficient POE BCE, hence statement ❶->S->AD If statement ❶ is insufficient POE AD, hence statement ❶->IS->BCE (D) Read statement ❷ alone and decide sufficient/insufficient. (E) Don't forget to POE as you go along. (F) Make sure you don't use words such as "yes" or "no" for they may add to the confusion that is already there.

Data Sufficiency - What does SUFFICIENT mean???

(a) A statement is sufficient if it gives you enough data to solve the question. However, you do not need to really solve it. (b) In Data Sufficiency treat each statement alone. Start with statement ❶. (C.) If statement ❶ is sufficient, POE answer choices B, C, and E, and keep answer choices A and D. Hence, Statement ❶->Sufficient->AD. (D) If statement ❶ is insufficient, POE answer choices A and D, and keep answer choices B, C, and E. Hence, Statement ❶->Insufficient->BCE.

Percents, Fractions, and Decimals - One Big Family???

Here are the rules for transforming decimals and fractions to percents and vice versa; ❶ To convert from fraction→percent, multiply by a 100 (e.g., 1/3·100=33.3%). ❷ To convert from decimal→percent, multiply by a 100 (e.g., 0.125·100=12.5%). ❸ To convert from percent→fraction, divide by a 100 (e.g., 25% = 25/100=1/4). ❹ To convert from percent→decimal, divide by a 100, moving the decimal point to the left (e.g., 16.6%:100=0.166) Memorize the following chart. It's a big time saver. That way you will be able to switch from decimals, to fractions, to percents. ❺ Fractional Values: ½ = 0.5, 1/3 = 0.33, 2/3 = 0.66, 1/4 = 0.25, 3/4 = 0.75, 1/5 = 0.2, 2/5 = 0.4, 3/5 = 0.6, 4/5 = 0.8, 1/6 = 0.166, 1/8 = 0.125, 1/9 = 0.111, 1/11 = 0.0909.

Sets: Table A/not A - How to Set a Table???

How to set a table: A/Not A Table: ❶ Draw an empty 3×3 table. ❷ Place one set (Set A) as the the title of the first row. ❸ Title the second row as the complementary set (Not A) and the third row as the Total. ❹ Repeat the process in the columns for the other set in the question. ❺ Insert the data in the stem question to the proper cells. ❻ Mark the cell that denotes the requested value with a question mark. ❼ Calculate the value of the empty cells based on the rule that every row/column in the table fulfills the summation X+Y=Z.

Powers: Exponential Equations???

Identifying exponential equations - an equation with variables in the exponents. Solving the problem (3-step process): ❶ Bring both sides of the equation to the same base(s). ❷ Ignore the bases and equate the exponents. ❸ Solve for the needed variables.

Integers: Factoring - The Factor Chart???

The factor chart is a basic technique for finding all the factors (prime or non-prime) of any integer: ❶ Write the original integer on top of the table. ❷ Divide the original Integer by the smallest possible factor: 1. Write "1" in the left column, and the result of the division in the right column. Continue dividing by "2", "3" etc., writing the results in the right column. Skip any factors that the original integer is not divisible by. ❸ Continue the process until the factors repeat themselves - stop when you're trying to divide the original integer by a factor that's already on the right side of the table. The table now holds the factors (prime and non prime) of the original integer. Example for 60: 1 X 60, 2 X 30, 3 X 20, 4 X 15, 5 X 12, 6 X 10.

If more than 1 answer is left standing in SNAPSHOT BALLPARK???

Choose whether to: ❶Proceed to finer POE considerations ❷Proceed to a direct geometric calculation

Sets: Complementary sets???

Complementary set: Complementary to set A: Is the set of all items in the sample space that are not included in set A. Mathematical phrasing: Not A. Everyday language phrasing: Everything but A. In special cases set B can be complementary to set A if everyday knowledge denotes that B is the only Not A that exists - boys/girls, smokers/non-smokers etc. Notice that opposites doesn't necessarily mean complementary: tall is not complementary to short, happy is not complementary to sad, etc. When it is stated in the question that every item in the sample space belongs to either set A or set B, and no other option exists, then for this question set B is complementary to set A.

Sequences: Consecutive Integers - Definition???

Consecutive Integers: integers that follow in sequence, each number being 1 more than the previous number. Examples: 1, 2, 3, 4...; -10, -9, -8... Consecutive Even Integers: A set of even integers with a distance of 2 between each member each member and the following/preceding member. Examples: 2, 4, 6... Consecutive Odd Integers: A set of Odd integers with a distance of 2 between each member each member and the following/preceding member. Examples: 1, 3, 5...

Sets: Union between Sets???

Counting every item that belongs to any of the sets is the way to get to the union of the sets. Union [A,B]: A U B all items that belong to set A or B (or both). The union is therefore comprised of : only A + only B + Intersection of A and B (both).

Averages: The Average Pie (a.k.a. The Plumber's Butt)???

(a) In GMAT averages problem solving questions, when any two parts of the average formula are given, you can and should find the third part. (b) In Data Sufficiency problems involving averages, two parts of the average formula are needed to determine sufficiency. (c) It is more convenient to use the average formula in the average pie format (or better yet, the plumber's butt). (d) The relations between the different parts of the formula are shown in the average pie. (e) Every time the word "average" appears in a problem, draw a plumber's butt. Don't forget to label the three areas to avoid confusion..

POE: Dogmatic Thinking???

(a) Test takers prefer positive numbers to zero or negatives. (b) Test takers prefer integers to fractions or decimals. (c.) Test takers prefer adding to subtracting numbers. (d) Test takers prefer multiplying to dividing. Keep these in mind, and try to avoid them. Think outside the Box!

Averages: Overview???

(a) The average of a set of numbers is a single value that describes the "center" value of the set. (b) The average is given in the formula: (Image not able to upload). (c.) The average of a set of number is not necessarily part of the set. Ex: The average of {3,7,8} is (3+7+8) ÷ 3=18 ÷ 3=6. Ex1: The average of {3,4,5} is (3+4+5)÷3= 12 ÷ 3=4. (d) Arithmetic mean, or just mean, has the same meaning as average.

Math Fundamentals: Integer Definition???

1) A real number is a fancy name for "number". When a GMAT question uses "real numbers", the number can be anything: positive, negative, integer or fraction. 2) An Integer is a non-fraction or a non-decimal. 3) By the above definition, zero is an integer.

Powers: Adding and Subtracting Powers???

1) Adding and subtracting powers with the same base: DON'T: add or subtract the exponents. Ex: x³ + x^5 ≠ x^8. Do: extract the greatest common factor. Ex: x³ + x^5 = x³(1+x²).

Integers: Rule of Divisibility by 3???

2Rule of divisibility by 3 - A number is divisible by 3 if the sum of its digits is divisible by 3.

Integers: Factor = Divisor???

A factor is a positive integer that divides evenly into another integer. A factor is a positive divisor. all of these GMAT question phrasings mean the same thing: ❶ 5 is a factor of x. ❷ 5 is a divisor of x ❸ x is divisible by 5 ❹ x÷5 = Integer.

Roots: Fractional Exponents - Conversion Between Root and Power Form???

A fractional exponent is another way of writing a root: The numerator of the fraction is the power. The denominator of the fraction is the root. Ex: See the image.

Math Fundamentals: Natural Numbers???

A natural number is simply a positive integer.

Powers: Overview???

A power is simply shorthand for "multiply the base by itself several times". e.g. x5 is simply a short way of writing x·x·x·x·x, or "x times itself 5 times". The x is called the "base". The 5 is called the "exponent". Four special powers: ❶ 423^1 = 423 ❷ 1^423 = 1 ❸ 423^0 = 1 ❹ 0^423 = 0

Math fundamentals: Prime Numbers???

A prime number is a natural number with exactly two distinct factors: 1 and itself. Remember the following facts about primes: ❶ 1 is not considered prime. ❷ 2 is the smallest prime. ❸ 2 is the only even prime.

Roots: Different Base, Same Root???

Arithmetic operations involving equal Roots with different bases: ❶ Multiplication / Division: Combine the two bases under the same root. Ex: √2·√8 = √(2·8) = √16 = 4. Ex1: √75 ÷ √3 = √(75 ÷ 3) = √25 = 5. Note: split complex bases into building blocks under the same roots. Look for building blocks with easily calculable roots, such as Perfect squares. Ex: √50 = √(25·2) = √25·√2 = 5·√2. Combining bases under different roots is an illogical concept. For example: √25 / ∛5 ≠ √(25/5). ❷ Addition / Subtraction: Do not add / subtract the bases. Beware of traps: Ex1: √2+√3 ≠ √5. Ex2: √7 - √4 ≠ √3.

Powers: Different Base, Same Exponent???

Arithmetic operations involving powers with different bases, same exponents: 1) Multiplication / Division: Combine the two bases under the same exponent. Ex1: 2³·3³ = (2·3)³ = 6³. Ex2: 15³ ÷ 3³ = (15 ÷ 3)³ = 5³. Note: split complex bases to building blocks under the same exponents. Ex3: 30² = (3·10)² = 3²·10² = 9·100 = 900. 2) Addition / Subtraction: Do not add / subtract the bases. Beware of traps: Ex1: 2²+3² ≠ 5². Ex2: 7² - 4² ≠ 3²

Plugging In: Avoiding Bad Numbers???

Bad numbers can make several answer choices match your Goal and waste your time. Therefore, it is best to avoid them: ❶ Don't use 0 or 1. ❷ Don't use numbers that appear in the problem or the answer choices. ❸ Don't use the same numbers for different variables. ❹ Don't use conversion numbers (numbers that convert between units. For example, 60 is the conversion number for minutes and hours). Instead, use double or half of the conversion number.

What is Ballparking???

Ballparking is one of the chief methods. Use approximate, easy-to-handle numbers to make the math easier and quicker. Arrive at a ballpark answer, and eliminate all answer choices that are not within the ballpark

Sets: Overview - What is a Set???

Definitions: Set = Group of objects. Sample space = All the relevant objects from which it is possible to select a set.

Math fundamentals: Even and Odd - Definitions???

Even: any integer that is divisible by 2 with no remainder. Examples: 2, 4, 14. Odd: Any integer that is not Even - not divisible by 2 with no remainder. Examples: 1, 3, 5, 7, 9. Remember: Zero is Even.

Combinations: Factorials- Basic Definitions???

Factorials are marked by a "!" sign. n! means "n multiplied by all consecutive integers less than n down to 1": n·(n-1)·(n-2)·...·1. For example: 5! = 5·4·3·2·1. Remember these special factorials: 0! = 1! = 1.

Integers: Factoring - Factors Vs. Multiples???

Factors and multiples are essentially opposite terms: Factors of a number are positive integers that the number divides into. Multiples of a number are formed by multiplying that number by any integer. All of these GMAT question phrasings mean the same thing: ❶ 5 is a factor of x. ❷ 5 is a divisor of x. ❸ x is divisible by 5. ❹ x/5 = Integer ❺ x = 5 X Integer. ❻ x is a multiple of 5. All of the above basically tell you that x=5, 10, 15, 20, 25...but also -5, -10, -15. [[continue]] One final note regarding zero: Zero is never a factor of another integer, as dividing by zero is not defined. Zero is a multiple of any integer. For example, zero is a multiple of five, as it is basically 5⋅0 - which is still 5 times an integer. Thus, if x is a multiple of 5, then x could still equal 0 - unless the problem indicates that x cannot equal zero (e.g. x is positive).

Integers: Questions Involving Fractions???

For integer problems involving fractions, break down the numerator and denominator into prime factor building blocks. Then, write down the expanded numerator and denominator on your noteboard, and ask yourself - "what does the problem really ask"?... If the result of the division is an integer, all of the denominator's building blocks must also be included in, and canceled by, the numerator's building blocks breakdown.

Integers: Factoring - Calculating the Number of Factors Using the Factor Chart???

For questions with large numbers that ask for the number of factors of an Integer: ❶ Test divisibility of the integer by all integers from 1 to the square root of the integer. Use the rules of divisibility to ask "Is this Integer divisible by 1? by 2? by 3?" etc. We'll discuss the Rules later. ❷ Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table. ❸ Count the number of small factors and multiply by 2. That's the number of factors. Ex: 140 = 1,2,4,5,7,10. Number of small factors: 6 [1, 2, 4, 5, 7, and 10]. 2 × Number of factors of 140: 6 × 2 = 12 factors.

Reverse Plugging In: What to Do with an Ugly Number in the Middle Answer?

In Reverse PI situations when there's an uncomfortable, "ugly" number in the middle answer: ❶ Don't force the middle answer into the problem. It may cost you time and possibly careless errors. ❷ Start by Reverse PI answer choices B or D instead, but never with the extremes (answer choices A and E) so you can POE effectively. ❸ Don't waste time trying to figure out which way to go. Pick a direction and go. ❹ Keep going as in all Reverse PI problems, until you find the answer that fits.

Data Sufficiency: Introduction???

In the Quantitative section, roughly 16 of the 37 questions will be in the Data Sufficiency (DS) question format. DS questions are made up of three parts: 1. The question stem. 2. Two statements. 3. Five answer choices. In DS questions, you are not trying to answer the question stem. Instead, you must determine whether the statements, either alone or together, are sufficient or insufficient to answer the question stem. The answer choices are always the same for all DS questions, covering all the possible combinations in sufficiency: (A) - Statement ❶ ALONE is sufficient, but statement ❷ alone is not sufficient. (B) - Statement ❷ ALONE is sufficient, but statement ❶ alone is not sufficient. (C) - BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient. (D) - EACH statement ALONE is sufficient. (E) - Statements ❶ and ❷ TOGETHER are not sufficient.

Sets: Intersection between Sets???

Intersection: - A ∩ B All items that belong to set A and to set B.

Quant is placed in which sections, Total Questions, Total Time, Scoring, Question Types???

It is the 3rd Section, 37 multiple choice questions, 75 minutes, 0-60 (effectively 7-51), Problem Solving, Data Sufficiency.

What is Ballparking Geometry: Trust the Diagram means?

It means we can trust the diagram and use the accurate figure to bypass complex calculations. In any (rare) exception a clear warning will be given: "Note: figure not drawn to scale".

Plugging In: Invisible Plugging In - Percents???

Let's review the basics of Hidden Plugging In - Percents: ❶ Invisible Plugging In problems are variable-free. ❷ Don't go looking for variables in the answer choices, instead spot the invisible variable in the problem. It may look like "the budget, the job, the shoes", or anything you can replace with a variable. ❸ If the question asks about percents, plug in 100 or a multiple of 100 (i.e. 200, 300, etc.). ❹ Substitute difficult words with simpler ones to understand what to do.

Plugging In: Must Be Questions???

Must Be Questions look for the one answer that is always true, for any number. The correct approach to these problems is to Plug in more than once and try to "break" the problem - try to find an example for which the question is NOT true. The basic steps for Must Be Questions: (a) Try to figure out the issue of the problem. That's how you'll know what to Plug In, especially, while plugging in more than once. Here are a few rules of thumb to get you going; (b) If the question contains multiplication and divisibility, try to plug in positive vs. negative numbers. (c) If the question contains exponents and roots, try to plug in integers vs. fractions. (d) Plug In good numbers first, then POE. (d) If several answers remain, ask yourself for each answer choice: Is it always true, for any number?. (e) Plug In again using DOZEN F, then POE. Keep Plugging In until only one choice remains. Remember the DOZEN F: (a) Different (e.g. even vs. odds, prime vs. multiple, fractions vs. integers, positive vs. negative, etc.). (b) One. (c) Zero. (d) Equal numbers for different variables. (e) Negatives. (f) Fractions. (g).

Powers: Basic Rules - Negative Exponents???

Negative exponents signify a reciprocal relation. Anything raised to a negative exponent becomes 1 ÷ [the original power without the negative sign]. Ex: a^−n = 1 / a^n

What is Ballparking Geometry - work order - start with a snapshot ballpark???

One must Always Start with a snapshot ballpark. ❶ Compare the Target value (length/area/angle) to a Given value in the question - how many times does one fit into the other? ❷Quickly eliminate out all answers that are clearly out of the ballpark.

Powers: Even and Odd Powers - Effect on Sign???

Remember this difference between even and odd powers: An even power is always positive, whether the base is positive or negative. An odd power retains the base's original sign. Remember to determine whether or not the negative sign is included in the base: -a^n ≠ (-a)^n

Roots: Overview???

Roots are simply the opposite of powers. A root asks a question: Which number, when raised to the root's power, will equal whatever is under the radical sign? A quadratic equation such as x^2=4 still has two solutions: in this case, both x=2 and x=-2 will satisfy the equation. BUT FOR EVEN ROOTS, where there are two possible answers for the above question (a positive and a negative) "ROOT" means "ONLY THE POSITIVE ROOT". Thus, √4=2.

Integers: Rule of Divisibility by 11???

Rule of divisibility by 11 - a number is divisible by 11 if the difference between the sum of its digits in the odd places and the sum of its digits in the even places is divisible by 11. In mathematical formula form: (sum of digits in odd places) - (sum of digits in even places) = a number that is divisible by 11.

Integers: Rule of Divisibility by 2???

Rule of divisibility by 2 - a number is divisible by 2 if its last digit (the unit's digit) is even.

Integers: Rule of Divisibility by 4???

Rule of divisibility by 4 - a number is divisible by 4 if its last two digits form a number that is divisible by 4.

Integers: Rule of Divisibility by 5???

Rule of divisibility by 5 - a number is divisible by 5 if its last digit is either 5 or 0.

Integers: Rule of Divisibility by 6???

Rule of divisibility by 6 - a number is divisible by 6 if it satisfies both of the following conditions: (a) Rule of divisibility by 2 - Its units digit is even. (b) Rule of divisibility by 3 - The sum of its digits is divisible by 3.

Integers: Rule of divisibility by 7???

Rule of divisibility by 7: 2·[# of hundreds]+[remaining number] should be divisible by 7.

Integers: Rule of Divisibility by 8???

Rule of divisibility by 8 - a number is divisible by 8 if its last 3 digits form a 3-digit number that is divisible by 8.

Integers: Rule of Divisibility by 9???

Rule of divisibility by 9 - a number is divisible by 9 if the sum of its digits is divisible by 9.

Integers: Rules of Divisibility Overview???

Rules of divisibility are quick ways to decide whether any number is divisible by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11.

Math Fundamentals: Arithmetic Operations - Definitions???

SUM - the result of addition (+). DIFFERENCE - the result of subtraction (-). PRODUCT - the result of multiplication (×). QUOTIENT - the result of division (:).

Powers: Scientific Notation???

Scientific notation: a×10^n where a (the digit term) indicates the number of significant figures in the number and 10^n (the exponential term) places the decimal point. ❶ A positive exponent shows that the decimal point is shifted that number of places to the right. ❷ A negative exponent shows that the decimal point is shifted that number of places to the left. The rule: Maintain the balance between the digit term and the exponential term. If one goes up (↑) by a magnitude of 10, the other must go down (↓) by the same magnitude, and vice versa.

Sets: Table A/B???

Set B is complementary to set A if the question defines that only sets A+B cover all of the sample space, and there is no intersection between A and B. If the question contains sets that are complementary, the set table is used with one modification - | A | Not A | Total | will be changed to | A | B | Total |.

Reverse Plugging In: Hard To Spot Reverse PI Situations - Basic and Additional Stop Signs???

The advanced Stop Signs are more apparent. However, the basic Stop Signs which are sometimes overlooked, are essential to define a Reverse PI situation. The basic Stop Signs for Reverse PI situations are: ❶ Invisible variable in the problem. ❷ Specific question. ❸ Numbers in the answer choices. The advanced Stop Signs for Reverse PI situations are: ❶ Feeling an algebraic urge to set up one big equation. ❷ The problem presents a long convoluted story. ❸ You feel stuck, not sure how to approach the problem.

Data Sufficiency: Plugging into Yes/No Data Sufficiency???

The basic steps for Yes/No Data Sufficiency Plugging In: ❶ Figure out the issue before you dive deeper into the statements. (The issue in Yes/No Data Sufficiency means which number(s) yield a Yes or a No). The issue is also an important step in solving Must Be questions. ❷ Regard the statements as facts that cannot be broken. Only then should you plug in the numbers you used in the question stem. ❸ Ask yourself every time you get a Yes or No: "Is it always true, for any number?" This is completely equivalent to Must Be questions. Remember: only a definite Yes/No is Sufficient. | Remember the DOZEN F: (a) Different (e.g. even vs. odds, prime vs. multiple, fractions vs. integers, positive vs. negative, etc.). (b) One (c) Zero (d) Equal numbers for different variables (e) Negatives (f) Fractions.

Integers: Factoring - Calculating the Number of Factors Using the Factor Chart - Perfect Squares???

The quick method for counting the number of factors of a perfect square remains the same, with a slight twist: ❶ Test divisibility of the integer by all integers from 1 to the square root of the integer. Use the rules of divisibility to ask "Is this Integer divisible by 1? by 2? by 3?" etc. Remember - we'll discuss the Rules later. ❷ Record the small factors (the numbers that the original integer IS divisible by) in the left side of the table. ❸ Count the number of small factors and multiply by 2. ❹ Subtract 1 from the total - the square root cannot be counted twice. The result is the number of factors.

Math Fundamentals: Divisible = Divisible Without a Remainder???

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Plugging In: Basic Technique???

To spot Plugging In problems: ❶ look for variables in the answer choices. ❷ look for the phrase "in terms of...". To Plug In, follow these steps: ❶ Substitute each variable in the problem with a number. Work the problem using the number(s) you chose until you come up with a number that answers the question. Circle it. That's your Goal. ❷ Plug in the same number into the same variables in the answer choices. Eliminate any answer that does not equal your Goal. ❸ Remember to check all five answer choices while plugging in. The right answer is right only because all other answer choices are eliminated, as they do not match your Goal. Remember: It is easier to master Plugging In with easy problems first, because the principles are the same for easy problems as for hard ones. Get into the habit now, and reap the rewards later.

Fractions: Comparing???

Use the Bowtie technique to compare fractions: ❶ Write down the fractions side by side. ❷ Draw two Bowtie arrows and multiply along each arrow ❸ The greatest fraction is the one with the greatest product.

Integers: Factoring - The Factor Tree???

Using the Factor tree to find the prime factors of an integer: ❶ Divide the original integer by the smallest possible prime number: 2. If the integer is not divisible by 2, try dividing by the next smallest prime: 3, then 5, etc. Write the prime on the left side of the tree, result of the division on the right. ❷ If the result is not prime - continue dividing by the smallest possible prime number, marking primes on the left, results on the right. ❸ Repeat the process until the result of the last division is prime - then stop. ❹ Circle the left-and-bottom-most integers on the tree - these are the prime factors of the original integer.

Roots: Basic Rules - Dividing Roots with the Same Base???

When Dividing roots with the same b, simply convert the root to power form and subtract the exponents. Ex: √a ÷ ∜a = a^½-¼ = a^¼ = ∜a

Powers: Basic Rules - Dividing Powers with the Same Base???

When dividing powers of the same base, subtract the exponents. a^n ÷ a^m = a^n-m

Powers: Reverse Rules - Dividing Powers with the same Base???

When dividing powers of the same base, subtract the exponents. a^n ÷ a^m = a^n-m. This rule may also be applied in reverse. To do so, first rewrite the exponent as a subtraction. Next, apply the parts of the subtraction as exponents in two powers divided by one another, each with the same base as the original power. Ex1: x^4 = x^7-3 = x^7 ÷ x³. Ex2: 7³x = 7^5x-2x = 7^5x ÷ 7²x. This is also the proper way to treat an exponent which already includes a subtraction sign: Ex1: xʸ-5 = xʸ ÷ x^5. Ex2: y^7m-2 = y^7m ÷ y². Ex3: 6ᵡ-1 = 6ᵡ ÷ 6^1 = 6ᵡ ÷ 6.

Powers: Reverse Rules - Multiplying Powers with the same Base???

When multiplying powers of the same base, add the exponents. Ex: a^n·a^m = a^n+m. This rule may also be applied in reverse. To do so, first rewrite the exponent as an addition. Next, apply the parts of the addition as exponents in two powers multiplied by one another, each with the same base as the original power. Ex1: x^4 = x³+1 = x³∙x. Ex2: x^7 = x^5+2 = x^5∙x². This is also the proper way to treat an exponent which already includes an addition sign: Ex1: xʸ+5 = xʸ∙x^5. Ex2: y³n+1 = y³n∙y. Ex3: 3^ᵡ+1 = 3^ᵡ∙3^1 = 3^ᵡ⋅3.

Powers: Basic Rules - Multiplying Powers with the Same Base???

When multiplying powers of the same base, add the exponents. a^n·a^m = a^n+m

Roots: Basic Rules - Multiplying Roots with the Same Base???

When multiplying roots with the same base, convert the root to power form and add the exponents. Ex: √a·∜a = a^½·a^¼ = a^½+¼ = a^¾ = ∜a³

Powers: Basic Rules - Raising a Power to Another Power???

When raising a power to another power, multiply the exponents. Ex: (a^m)^n = (a^n)^m = a^m·n. The equation works both ways: a^6 can be split up to a³⋅2, which can then be rewritten as the equivalent (a³)² or (a²)³, as needed by the question.

Powers: Reverse Rules - Raising a Power to Another Power???

When raising a power to another power, multiply the exponents. Ex: (a^m)^n = (a^n)^m = a^m·n. This can also be done in reverse. To do so, split the exponent into a product, and then 'pull' one of the multiplicands into a pair of brackets. Ex: a^6 can be split up to a³⋅2, which can then be rewritten as the equivalent (a³)² or (a²)³, as needed by the question.

Roots: Basic Rules - Raising a Root to a Power???

When raising a root to another power, simply convert the root to power form and multiply the exponents. Ex: (√a)² = (a^½)² = a^½^·2 = a^1 = a. roots and powers of the same level cancel each other out, leaving only the base. Ex1: ∜a^4 = a.

Data Sufficiency: The Question Stem - What is the Type???

When solving a DS question you must first extract all possible data from the question stem: ❶ Type: Determine whether it is a value question or a yes/no question. ❷ Issue: Figure out the GMAT knowledge field that is tested (e.g averages, quadrilateral area formulas etc.). ❸ Missing piece: While focusing on the question stem, determine what is the piece of information that's needed to answer the question asked. Always try to figure out the issue of the stem before you approach the statements. This way, you'll be more focused on what you are looking for, thereby determining whether the statement is Sufficient or Insufficient more efficiently.

Plugging In: Invisible Plugging In - Fractions???

❶ In Hidden Plugging In problems the variables aren't visible but rather hidden. ❷ Spot the invisible variable in the problem, e.g. "the budget", "the job", or "the shoes", and then solve by plugging in. ❸ If the question asks about fractions, multiply the bottoms to find a good number to plug in.

Math Fundamentals: The number Line - Left and Right of Zero???

❶ Positive: >0 ❷ Negative: <0 ❸ Zero is neutral - neither positive nor negative.

Math Fundamentals: Non-positive and Non-negative???

❶ Positive: >0 ❷ Non-negative: ≥0 ❸ Negative: <0 ❹ Non-positive: ≤0

Ballparking Geometry: Useful Numbers to Remember???

∏ = 3+(Little more that 3), √2 = 1.4, √3 is approximately 1.7

Fractions: Overview???

❶ A fraction is defined as part/whole. ❷ In the fraction 3/4, 3 is the numerator, and 4 is the denominator. ❸ The fraction bar (the line that separates the numerator from the denominator) denotes division (÷). ❹ An improper fraction is one in which the number on top (part) is greater than the bottom (whole). Treat it as if it were a simple fraction.

Math Fundamentals: Quotient, Divisor, Dividend???

❶ A quotient is the result of a division between two numbers. ❷ Dividend÷Divisor=Quotient. ❸ When dividing two integers, the quotient refers only to the integer part of the result. (not the remainder)

Integers: Prime factors are Building Blocks???

❶ An integer n is divisible by a smaller integer g if n is divisible by all of g's distinct prime factors. ❷ When testing divisibility using larger building blocks that are not prime, remember to use building blocks that have no common divisor greater than 1.

DS: Yes/No Basic Technique???

❶ Answering a definite "Yes" or a definite "No" means Sufficient. ❷ If the answer is sometimes "Yes" and sometimes "No", it means Maybe, which means Insufficient. The only way a statement will be insufficient is if it allows both a "Yes" and a "No" answer. ❸ After that, follow the Data Sufficiency FlowChart to get the final answer.

Integers: Finding the LCM - Least Common Multiple???

❶ Break down each integer into its "building blocks" using the factor tree. ❷ Build a list comprised of the least number of prime "building blocks" required to "build" each of the integers. ❸ Multiply the building blocks in the list to find the LCM. Remember that each prime number is included the minimum number of times required to build the LCM. the LCM of 12, 27 and 36 is 108.

Fractions: Reducing & Expanding???

❶ Expanding a fraction means multiplying the top and bottom by the same number (i.e., expanding 2/3 by 2 yields 4/6) ❷ Reducing a fraction means dividing the top and bottom by the same number (i.e., reducing 6/63 by 3 yields 2/21). ❸ Expanding or reducing doesn't change the relationship between the numerator and denominator. ❹ To get a fraction to its most reduced form, keep reducing it until you can't find a number that divides into both numerator and denominator.

Reverse Plugging In: Basic Technique???

❶ Identify what the question is asking. Reverse Plugging In questions are always specific (i.e. how much?, how many?, what is the number of...?, what is the value of x? etc.) ❷ The two major identifiers of Reverse PI questions are a specific question and numbers in the answer choices. ❸ Plug the middle answer (C) back into the question: Work the problem assuming that the answer is C. If everything checks out - that's the answer. If not:. ❹ Notice that numerical answer choices are always in ascending/descending order. Figure out in which direction to move on so you are able to POE effectively. ❺ There is no need to check all five answer choices as you must do in "regular" Plugging In questions. If you find an answer that fits the problem - stop. Pick it...

Reverse Plugging In: Not Sure In Which Direction To Go???

❶ If the answer you Plugged in doesn't fit, POE it, and try to decide in which direction to go. ❷ If you can't make up your mind in which direction to go after 10 seconds, pick a direction and go. ❸ Some answer choices do not need to be plugged in - Try to POE answer choices that don't fit the problem before you Reverse PI. ❹ The Reverse PI process within the question becomes faster and faster as you advance down the answer choices and embed the logical path in your mind.

Math Fundamentals: The Remainder???

❶ Remainder is the distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend. ❷ The greatest possible remainder is one less than the divisor. e.g. When dividing by 5, the highest possible remainder is 4. ❸ Identifying remainder problems in the GMAT - the question uses the word remainder (Duh!). ❹ GMAT problems involving remainders can usually be easily solved by Plugging in numbers that fit the problem. Try plugging in the remainders themselves: When X is divided by 5, the remainder is 3 - plug in X=3. ❺ when plugging in is difficult to use, use the following formula: for any integer i divided by another integer d. Form: i = quotient·d + remainder. Thus, if dividing i by 5 leaves a remainder of 3, i can be expressed as the equation: Form: i=5x+3. x being the quotient.

Fractions: Reciprocal???

❶ The reciprocal or the inverse of a fraction is that fraction flipped over; top to bottom and bottom to top. i.e., the reciprocal of 2/5 is 5/2. ❷ Remember that the reciprocal doesn't change signs. ❸ To find the reciprocal of an integer, write it in fraction form first, then flip. i.e. 5 can be written as 5/1, hence the reciprocal is 1/5..

Fractions: Adding and Subtracting???

❶ To add/subtract fractions with the same denominators (bottoms), simply add/subtract the numerators (ups). ❷ To quickly add/subtract fractions with different denominators use the bowtie. For example, 1/6 + 1/8 will work out as:(Image not able to add).

Fractions: Dividing???

❶ To divide one fraction by the other, find the reciprocal of the second fraction (the divisor), then multiply. ❷ Remember: Dividing by a fraction is the same as multiplying by its reciprocal. ❸ When dividing a fraction by an integer, first write the integer as a fraction, then divide.

Fractions: Multiplying???

❶ To multiply fractions, multiply straight across, tops with tops, bottoms with bottoms. ❷ If possible reduce before you multiply. ❸ When multiplying a fraction and an integer, first write the integer as a fraction, then multiply.

PI: Using Good Numbers???

❶ Use good numbers (small, positive integers) while Plugging In to make the math easy and error-free. ❷ If the question asks about dozens of eggs, choose 24, 36, etc. If the question asks about a number that's divisible by 15, choose a multiple of 15, etc. ❸ If the question asks about fractions, choose a number that's a multiple of the denominators (the best way to go about it is simply to multiply the bottoms of the fractions). ❹ If the question asks about percents, choose 100 or multiples of it.

Plugging In: Overview???

❶ When GMAC messes around with different units in the same problem be super-extra careful (e.g. Dollars vs. Cents). ❷ Don't try to approach algebraic problems with variables in an algebraic way. Use numbers instead. That way you can be 150% sure with your answer choices. In other words, use Plugging In when variables are around.

Integers: Integer 'Must be True' Questions???

❶ When asked which of the following n MUST be divisible by, come down to the minimum number of building blocks that you know for sure n MUST be divisible by. Everything not in the list of minimum building blocks falls under "CAN" - not "MUST". ❷ The prime factors of any integer that is a power of another integer come in pairs, triplets, quadruplets etc. according to the power. If a is an integer, and a^2 is an integer, then a^2's prime factors must come in pairs.

4 Sections of GMAT???

❶Analytical Writing Assessment (AWA)\ 1 Argument essay \ 30 minutes. ❷Integrated Reasoning \ 12 questions on four Question types \ 30 minutes *❸Quantitative section \ 37 questions \ 75 minutes ❹Verbal section \ 41 questions \ 75 minutes.

Things to Remember for Adaptive Scoring???

❶Make a strong first impression (roughly the first 10 questions of each adaptive section). ❷Never leave questions unanswered. ❸Avoid guessing the last bunch of questions. ❹Avoid making several mistakes in a row - this decreases your score more than making the same number of mistakes interspersed throughout the exam.