Odd and Even Numbers

1. Definition:

• Even Numbers: These are integers divisible by 2 without a remainder. They can be expressed as (2k), where (k) is an integer. Examples include -4, 0, 2, 4, 6, etc.
• Odd Numbers: These are integers that are not divisible by 2. They can be expressed as (2k + 1), where (k) is an integer. Examples include -3, 1, 3, 5, 7, etc.

2. Properties:

• Addition/Subtraction:

• Even ± Even = Even
• Odd ± Odd = Even
• Even ± Odd = Odd
• Multiplication:

• Even × Even = Even
• Odd × Odd = Odd
• Even × Odd = Even

3. Division:

• The division of even by even or odd by odd does not always result in an integer.
• Division of even by odd or odd by even can result in fractions.

4. Special Cases:

• Zero (0): Zero is considered an even number because it is divisible by 2.
• Negative Numbers: The rules for odd and even numbers also apply to negative integers.

5. Practical Applications:

• GRE Problems: You might encounter problems requiring you to identify patterns, solve equations, or determine the parity (odd/even nature) of results. Understanding these properties helps in quickly solving such problems.

Absolutely! Here are some GRE-style practice questions focused on odd and even numbers:

Practice Questions

1. If ( x ) and ( y ) are integers such that ( x ) is even and ( y ) is odd, which of the following must be true?

• A) ( x + y ) is even
• B) ( x y ) is odd
• C) ( x – y ) is odd
• D) (  ) is an integer

2. Which of the following expressions always results in an even number?

• A) ( 2a + 3b ) where ( a ) and ( b ) are integers
• B) ( a² + b² ) where ( a ) and ( b ) are odd integers
• C) ( 2a + 2b ) where ( a ) and ( b ) are integers
• D) ( a² – b² ) where ( a ) and ( b ) are even integers

3. If ( n ) is an odd integer, which of the following is true about ( n^2 )?

• A) ( n^2 ) is even
• B) ( n^2 ) is odd
• C) ( n^2 ) is divisible by 4
• D) ( n^2 ) is divisible by 2

4. Given that ( a ) and ( b ) are even integers, which of the following statements is true?

• A) ( a + b ) is odd
• B) ( a  b ) is odd
• C) ( a – b ) is even
• D) (   ) is always an integer

5. If ( x ) is an even integer and ( y ) is an odd integer, which of the following is always true?

• A) ( x² + y² ) is even
• B) ( x² – y² ) is odd
• C) ( x²  y² ) is odd
• D) ( ) is an integer

Answers:

B) ( x² – y² ) is odd

C) ( x – y ) is odd

C) ( 2a + 2b ) where ( a ) and ( b ) are integers

B) ( n² ) is odd

C) ( a – b ) is even