Free GRE Course > CONSECUTIVE NUMBERS

Note: this section is designed as a math review for students scoring in the low to middle ranges.

### Number Definitions

#### Integers

• a number such as 0, 1, 2, and 3 that has no fractional part.
• Integers can be positive {1, 2, 3, …}, negative {-1, -2, -3, …}, or zero {0}.
• The GRE will define whole numbers as positive integers and zero {0, 1, 2, 3, …}.
• In other sources the term “whole numbers” may refer to all integers or just positive integers.

#### Positive number

• a number greater than zero, such as +5 (usually written simply as 5).

#### Negative number

• a number less than zero, such as -5.

#### Zero

• neither positive nor negative, zero is an integer and an even number

#### Rational number

• a number that can be expressed as a fraction of two integers.

#### Irrational number

• a number, such as π or √2, that cannot be expressed as a fraction of two integers.

#### Division and Zero

• Addition, subtraction, and multiplication of integers will always result in an integer. However, division is different. Sometimes a quotient is not an integer, though it is a rational number:
• 12 ÷ 4 = 3, but 4 ÷ 12 = 1/3
• The result of a number divided by zero is undefined. For example, 8/0 = ? since there is no number 0 × ? = 8. Zero divided by a number is equal to zero. For example, 0/3 = 0 since 0 × 3 = 0.

#### Variable

• a letter used to represent one or more numbers. A variable expression has numbers, variables, and operations

If you forget a rule (which happens all the time under test day pressure) consider using the strategies in The 7 Techniques chapter, like Experiments or Backsolving/Plug-In.

### Order of Operations (PEMDAS)

Video Courtesy of Kaplan GRE

Take a look at the equation 5 + 2 × 3. Is the correct answer 21 or 11?

• (5 + 2) × 3 = (7) × 3 = 21 (add first, then multiply?)
• 5 + (2 × 3 ) = 5 + (6) = 11 (multiply first, then add?)

To address this issue you need to know the “Order of Operations,” which is the priority list for calculations.

The correct order of operations is Parentheses – Exponents – Multiplication – Division – Addition – Subtraction. An easy way to remember the order is by the initials PEMDAS or the mnemonic phrase Please Excuse My Dear Aunt Sally.

#### Rule 1: Please Excuse My Dear Aunt Sally

Parentheses

(-5 + 2)(-3) = (-3)(-3) = 9

Excuse

Exponents

22 + 4 = 4 + 4 = 8

My

Multiplication

5 + 2 × 3 = 5 + 6 = 11

Dear

Division

8 ÷ 2 + 2 = 4 + 2 = 6

Aunt

4 + 6 – 10 = 10 – 10 = 0

Sally

Subtraction

7 – 5 – 2 = 2 – 2 = 0

Parentheses can be used for notating fractions and negative numbers.
One-half can be written as 1/2 or (1/2).

• 25 × (1/5) = 5
• 5 + (-2) = 3

#### Rule 2: Combine all like terms in an expression.

The expression 2x + x – y + 4y can be simplified by combining the like terms (x‘s and y‘s) into 3x + 3y.

#### Rule 3: Distribute numbers to eliminate parentheses.

Note: Since subtraction is the same as addition of a negative number, e.g. 5 – 3 = 5 + (-3) = -2, these operations can be calculated interchangeably. Left to right is not always the best order.

Example: 15 + 8 – 6 – 5 is easier when you do 15 – 5 + 8 – 6 = 10 + 8 – 6 = 10 + 2 = 12.

Note: Multiplication and division are also interchangeable.

Example: 18 × 2 ÷ 3 ÷ 2 is easier when you do 18 × 2 ÷ 2 ÷ 3 = 18 × 1 ÷ 3 = 18 ÷ 3 = 6

• 2(x-y) = 2x – 2y
• -3(x-y) = -3x + 3y

Be sure to distribute the negative to each term!

#### Rule 4: Evaluate expressions' grouping symbols from the inside out.

• x(x + 2(3x + 4) – 3) …Distribute 2 into the inner parentheses: 2(3x+4) = 6x+8
• x(x + 6x + 8 – 3) …Combine the like terms (the x’s) inside the parentheses
• x(7x + 5) … Distribute to remove the last parentheses.
• 7x2 + 5x

#### Example

16 ÷ (2(8 – 3(4 – 2))) + 3 =

### Solution

16 ÷ 2(8 – 3(4 – 2)) + 3 = ?
Combine the inner parentheses: (4 – 2) = 2

16 ÷ 2(8 – 3(2)) + 3
Combine the next parentheses: 8 – 3(2) = 2

16 ÷ 2(2) + 3
Multiplication before division: multiply out 2(2) = 4

16 ÷ 4 + 3
Division, 16 ÷ 4 = 4 before addition

4 + 3 = 7

#### Example

10 – 3(8 – 3)2 ÷ 15 =

### Solution

10 – 3(8 – 3)2 ÷ 15 …Combine inside the parentheses

10 – 3(5)2 ÷ 15 …Calculate the exponent

10 – 3(25) ÷ 15 …Multiply

10 – 75 ÷ 15 …Divide

10 – 5 = 5

#### Example

|-2x2| – (3x/2)2 × 2

### Solution

| -2x2 | – (3x/2)2 × 2

2x2 – 9x2/4 × 2       …Do operations inside grouping symbols: absolute value and parentheses.

2x2 – 9x2/2       …Multiply.

4x2/2 – 9x2/2       …Use the common denominator and subtract.

-5x2/2

### Odd / Even Numbers

#### Even number

• an integer that is divisible by 2 (…, -6, -4, -2, 0, 2, 4, 6, 8, 10, …).
• No, don’t email us saying that’s a typo; zero is an even integer.

#### Odd number

• an integer not divisible by 2 (…, -5, -3, -1, 1, 3, 5, 7, 9, …).

#### Prime number

• a positive integer with exactly two different positive divisors: 1 and itself. For example, 2, 3, 5, 7 are prime numbers.
• Note: 1 is not a prime number since it has only one positive divisor.

Only integers can be odd, even, or prime numbers.

#### Adding, Subtracting, and Multiplying Odd, Even, and Prime Numbers

On the GRE, you may be asked to add, subtract, or multiply very large numbers in order to answer a relatively simple question. These rules can provide a shortcut through questions like that:

Rule
even + even = even
odd + even = odd
odd + odd = even
even – even = even
even – odd = odd
odd – odd = even
even × even = even
even × odd = even
odd × odd = odd

Example
4 + 4 = 8 (even)
3 + 4 = 7(odd)
3 + 3 = 6 (even)
16 – 8 = 8 (even)
16 – 5 = 11 (odd)
9 – 5 = 4 (even)
2 × 4 = 8 (even)
2 × 3 = 6 (even)
3 × 5 = 15 (odd)

#### Example

If k is an odd integer, are the following expressions even or odd?

• k + k + k …(k + k) is even. Thus (k + k) + k is an even plus an odd, which is odd.
• k × k × k …(k × k) is odd. Thus, (k × k) × k is an odd times an odd, which is odd.
• k + 2kk + 2k is an odd plus an even, which is odd.
• 2k × k …2k is even. An even times an odd is even.

examples:

Dividing even by even can be even, odd, or not an integer.

40 ÷ 4 = 10      40 ÷ 8 = 5      8 ÷ 40 = 1/5

Dividing even by odd can be even, odd, or not an integer.

30 ÷ 5 = 6      30 ÷ 6 = 5      15 ÷ 30 = 1/2

Dividing odd by odd cannot be even. 15 ÷ 3 = 5
Dividing odd by even cannot be an integer. 15 ÷ 2 = 7.5

#### Positive and Negative Numbers

All numbers, except zero, are positive or negative.

Adding two numbers can give a positive or a negative number.

• positive + positive = positive
• negative + negative = negative

Positive + negative can give a positive, a negative, or zero.

• -7 + 9 = 2
• 4 + (-5) = 4 – 5 = -1
• 2 + (-2) = 0

Subtracting two numbers can give a positive, a negative, or zero.

• positive – positive 9 – 7 = 2 … 7 – 9 = -2 … 7 – 7 = 0
• negative – negative -9 – (-7) = -2 … -7 – 9 = -16 … -9 – (-9) = 0
• positive – negative 9 – (-7) = 9 + 7 = 16
• negative – positive -9 – 7 = -16

Be sure to distribute the negative sign to all terms inside parentheses.

• 9 – (14 – 8) = 9 – 14 + 8 = 3
• 10 – (5 + 2) = 10 – 5 – 2 = 3

You can use the order of operations to check:

• 9 – (14 – 8) = 9 – 6 = 3
• 10 – (7) = 3

### Multiplication and Division

When multiplying or dividing: if the two numbers have the same sign, then the product is positive. If the two numbers have different signs, the product is negative. A way to remember this is Same Signs product PoSitive.

positive × positive = positive

positive × negative = negative

negative × negative = positive

6 × 2 = 12

6 × (-2) = -12 or (-6) × 2 = -12

(-6) × (-2) = 12

positive ÷ positive = positive

positive ÷ negative = negative

negative ÷ negative = positive

6 ÷ 2 = 3

6 ÷ (-2) = -3 or (-6) ÷ 2 = -3

(-6) ÷ (-2) = 3

#### 800score Strategies

Use Plug-In or Backsolving on odd/even and positive/negative questions. Make up numbers to substitute in for variables on questions that ask if variables result in a positive or negative number. For example, if x < 0, is x2 positive? Substitute -2 for x and you can see that (-2)2 is positive. This is also very useful for double checking.

Make a list. If a positive, negative or prime number question asks “How many two digit prime numbers” or “How many two digit numbers less than 50” then you have a finite (small) number of possibilities. Here it is useful to make a list of all the number possibilities rather than try to solve the question using algebra.

#### Absolute Value

The absolute value of a number, written as | -5 |, is the distance of a number from zero on the number line: | +5 | = | -5 | = 5.

The value of an expression inside the absolute value signs could be positive or negative.
If | x | = 7, then x could be 7 or -7.

This means that absolute value equations usually have two possible solutions.

#### Example

Solve |x + 3| = 8

### Solution

x + 3 = 8  or  x + 3 = -8 …Rewrite as two equations

x = 5  or x = -11 …Two solutions

#### Powers

Note: Powers are covered more completely in Section 6, Exponents.

Writing a number raised to a power is a way of expressing multiplication of a number by itself. A power has two parts, the exponent and the base. The exponent says the number of times the base is used as a factor.

#### Example

• 64 = 6 × 6 × 6 × 6 = 1,296 (base 6, exponent 4)
• 52 = 5 × 5 = 25 (base 5, exponent 2)
• 43 = 4 ×4 ×4 = 64 (base 4, exponent 3)

The expression 64 is read as “six to the fourth.”

Two powers have special names. Powers with the exponent 2 are called “squared,” since it can imply area. The expression 52 can be read as “five to the second power” or “five squared.”

Powers with the exponent 3 are called “cubed,” since it can imply volume. The expression 43 can be read as “four to the third power” or “four cubed.”

#### Powers of negative numbers

When using exponents with negative numbers, make sure your answer has the right sign. Pay careful attention to whether a negative sign is inside or outside of parentheses.

#### Example

• (-4)3 = (-4) × (-4) × (-4) = -64
• (-5)2 = (-5) × (-5) = 25 but -52 = -(5 × 5) = -25

#### Roots

Just like addition and subtraction “undo” each other, so do exponents and roots. You take a root to answer questions like:

If 3√64 = x, what is the value of x?

You know 4 × 4 × 4 = 43 = 64, so 3√64 = 4.

The expression √a is called a root or radical. The symbol √ is the radical symbol.

Radical symbols can have a number in front in the radical. The 3 in 3√64 is asking for the cube root. For square roots, the 2 is left off.

#### Example

• 3√8 = 2 since 23 = 8
• 3√(-1) = -1 since (-1)3 = -1

#### Odd and Even Roots

When taking an odd root, there is only one answer.

#### Example

• 3√125 = 3√(5 × 5 × 5) = 5
• 3√-125 = 3√((-5) × (-5) × (-5)) = -5
• 5√32 = 5√(2 × 2 × 2 × 2 × 2) = 2

When taking an even root, there will be two answers. But the sign √ denotes only the non-negative answer. For example, there are two square roots of 25: 5 and -5, since 52 = 25 and (-5)2 = 25. But √25 has only one value: 5, √16 = 4, √1 = 1, etc.

#### Example

• √16 = √(4 × 4) = 4
• 3√1 = 3√(1 × 1 × 1) = 1
• 4√16 = 4√(2 × 2 × 2 × 2) = 2

Be careful when dealing with equations – remember that an even root is undefined for negative numbers. A root must be positive, so √(x2) = |x|.

#### Example

If x2 = 4, what is x?

### Solution

Since 22 = 4 and (-2)2 = 4, this answer is x = 2 and x = -2.

##### Alternative solution
• x2 = 4
• √(x2) = √4
• |x| = 2
• x = 2 and x = -2 are the solutions of the equation.