Math Fundamentals

Question 1

Math Fundamentals: Quotient, Divisor, Dividend

Answer 1

A quotient is the result of a division between two numbers.

Dividend÷Divisor=Quotient.

When dividing two integers, the quotient refers only to the integer part of the result. (not the remainder)

Question 2

Zero is a multiple

Answer 2

Yes, zero is indeed a multiple of both a and b. In fact, zero is a multiple of every number.

A certain number ‘x’ is a multiple of another number ‘y’, as long as it’s possible to multiply y by some integer and receive x as the result. For example, 6 is a multiple of 3 because it’s possible to multiply 3 by some integer (2, in this case) and receive 6 as the result. 36 is a multiple of 4 because it’s possible to multiply 4 by some integer (9, in this case) and receive 36 as the result.

Now, let’s apply the same logic on zero. Zero is a multiple of x if it is possible to multiply x by some integer and receive zero as the result. Regardless of the value of x, it’s possible to multiply x by zero (which is an integer) and receive zero as the result. Therefore, zero is a multiple of every number.

Question 3

Integers: Adding and Subtracting Multiples

Answer 3

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

Question 4

Integers: Prime factors are Building Blocks

Answer 4

Stop Sign: A plural subject made up of singular nouns that are connected by and

A subject that is made of several singular nouns connected by and is plural and should be followed by a plural verb. This does not apply to other connectors (as well as, along with, etc.).

Incorrect: John and Jane is lovers.
Correct: John and Jane are lovers.

Question 5

Math Fundamentals: The Remainder

Answer 5

Remainder is the distance (in units) from the dividend to the nearest multiple of the divisor that is smaller than the dividend.

The greatest possible remainder is one less than the divisor. e.g. When dividing by 5, the highest possible remainder is 4.

Identifying remainder problems in the GMAT - the question uses the word remainder (Duh!).
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.
when plugging in is difficult to use, use the following formula: for any integer i divided by another integer d

i = quotient·d + remainder

Thus, if dividing i by 5 leaves a remainder of 3, i can be expressed as the equation:

i=5x+3

x being the quotient.

Question 6

Integers: Even and Odd - No Rules for Division

Answer 6

When dividing Even and Odd numbers, there are no rules. The result could be even, odd or even a fraction - depending on the numbers.

Question 7

PI: Using Good Numbers

Answer 7

Use good numbers (small, positive integers) while Plugging In to make the math easy and error-free.
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.
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)
If the question asks about percents, choose 100 or multiples of it.

Question 8

Integers: GCD - Greatest Common Divisor

Answer 8

Common divisors - definition: Two non-zero integers have a common divisor if they are both divisible by the same positive integer.

Remember this about the Greatest Common Divisor of two integers:

Two primes will always have a G.C.D of 1.
A G.C.D of 1 does not mean the two integers are prime - just that they have no common divisor greater than 1. Example: 8 and 9 have a G.C.D of 1.
An integer can serve as the G.C.D of itself and another integer. Example: 6 and 12 have a G.C.D of 6.

Question 9

Integers: GCD - Greatest Common Divisor

Answer 9

Common divisors - definition: Two non-zero integers have a common divisor if they are both divisible by the same positive integer.

Remember this about the Greatest Common Divisor of two integers:

Two primes will always have a G.C.D of 1.
A G.C.D of 1 does not mean the two integers are prime - just that they have no common divisor greater than 1. Example: 8 and 9 have a G.C.D of 1.
An integer can serve as the G.C.D of itself and another integer. Example: 6 and 12 have a G.C.D of 6.

Question 10

Math Fundamentals: Integer Definition

Answer 10

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.

Question 11

Integers: Factoring - The Factor Tree

Answer 11

Using the Factor tree to find the prime factors of an integer:

1) 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.
2) If the result is not prime - continue dividing by the smallest possible prime number, marking primes on the left, results on the right.
3) Repeat the process until the result of the last division is prime - then stop.
4) Circle the left-and-bottom-most integers on the tree - these are the prime factors of the original integer.

Question 12

Reverse Plugging In: Basic Technique

Answer 12

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.)
The two major identifiers of Reverse PI questions are a specific question and numbers in the answer choices.
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:
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.

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.

Question 13

Integers: Even and Odd - Rules of Multiplication

Answer 13

Even × Even = Even (e.g. 2×2=4)
Even × Odd = Even (e.g. 2×3=6)
Odd × Odd = Odd (e.g. 1×1=1)

Question 15

Integers: Even and Odd - Rules of Addition and Subtraction

Answer 15

Even ± Even = Even (e.g. 2+2=4; 4-2=2)
Even ± Odd = Odd (e.g. 2+1=3; 2-1=1)
Odd ± Odd = Even (e.g. 1+1=2; 3-1=2)

Question 16

Integers: Integer ‘Must be True’ Questions

Answer 16

1) 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”.
2) 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 a2 is an integer, then a2’s prime factors must come in pairs.

Question 17

Special Quadratics

Answer 17

Memorize these recycled quadratics:

I (a+b)2 = a2+2ab+b2

II (a-b)2 = a2-2ab+b2

III (a+b)(a-b) = a2 - b2

We’ll discuss how to identify them and what to do when you do in the near future.

Question 18

Quadratics: Factoring - The Solutions Formula - Finding The Number of Solutions

Answer 18

To find the solutions of a any quadratic of the form ax2+bx+c=0:

1) Identify the coefficients a, b and c (write them down in your notebook for good measure).
2) Plug the coefficients into The Solutions formula:
3) The two solutions are:

and

To find the number of solutions to a quadratic of the form ax2+bx+c=0, plug the coefficients into b2-4ac (the discriminant):

b2-4ac > 0 - 2 different solutions.

b2-4ac = 0 - one solution.

b2-4ac < 0 - no (real) solutions.

Question 19

Rate Problems: The Rate Equation

Answer 19

Rate is the work that is done in a unit of time.

It is calculated as R=W/T

When solving a question, we translate it into the Work table.

Rate

Time Work

The table represents the formula Rate×Time=Work so that the multiple of the two left columns

equals the right column.

1. Assign a value from the question to each appropriate cell in the work table.
2. Calculate the value that of the empty cell based on the equation.

Make sure you pay attention to units - do they match? Can they help you eliminate answer choices?

Question 20

Quadrilaterals: Rhombus

Answer 20

A Rhombus is a Parallelogram with all sides equal.

The diagonals of a Rhombus are perpendicular to each other, as well as bisect each other and their respective base angles.

Question 21

Quadrilaterals: Evolution

Answer 21

Quadrilaterals: Evolution

A quadrilateral may also be a parallelogram, or a rectangle, or a square.
A parallelogram may also be a rectangle, or a square.
A rectangle may also be a square.
A square is a regular quadrilateral.

Question 22

Triangles: Recycled Right Triangles - The 45:45:90 Triangle - Practical Use for a Given Hypotenuse

Answer 22

If b is the hypotenuse of a 45°:45°:90° triangle use this ratio box to get you going:

b = 90

(b |2)/2 = 45

Each term in the box is the measure of the side opposite the angle.

Question 23

Triangles: Recycled Right Triangles - The 45°:45°:90° triangle - Practical Use for a Given Leg

Answer 23

Triangles: Recycled Right Triangles - The 45°:45°:90° triangle - Practical Use for a Given Leg

If a is one of the legs of a 45°:45°:90° triangle use this ratio box to get you going:
45°45° 90°

a a a√2

Each term in the box is the measure of the side opposite the angle.

Question 24

Triangles: Recycled Right Triangles - The 30°:60°:90° Triangle Practical Use

Answer 24

Triangles: Recycled Right Triangles - The 30°:60°:90° Triangle Practical Use

If a is the smallest leg of a 30°:60°:90° triangle use this ratio box to get you going:
30° = a

60° = a√3

90° = 2a

Each term in the box is the measure of the side opposite the respective angle.

Question 25

Special Recycled Right Triangles

Answer 25

The special recycled right triangles are
the 30°:60°:90° triangle
the 45°:45°:90° or isosceles right triangle
These recycled triangles are termed special because they are identifiable by their
angles
representation in a figure (half of a square or an equilateral triangle, respectively).

Question 26

Triangles: Equilateral Area Formula

Answer 26

Triangles: Equilateral Area Formula

Save time by using the unique area formula for an equilateral triangle of side a: .
You can also use AREA= ½×BASE×HEIGHT, which works for any triangle.

Question 27

recycled triangles

Answer 27

Save time, needless calculations, and careless errors by using the recycled right triangles.
Memorize the simple recycled right triangles 3:4:5, 5:12:13, 8:15:17

Question 28

Lines & Angles: The Number of Diagonals in a Polygon

Answer 28

Lines & Angles: The Number of Diagonals in a Polygon

A diagonal is a line that joins two corners which aren’t already connected by a side.
The number of diagonals of a polygon is given in the formula:

n(n-3)/2

, where n is the number of sides.