While the positive or negative classification can apply to integers and decimals/ fractions, odd or even is a classification that only applies to integers.
Even integers are integers that are divisible by 2 i.e. when an even number is divided by 2, there is a zero remainder. All even integers must end in 0, 2, 4, 6, or 8 i.e. their unit’s digit must 0, 2, 4, 6 or 8. An even integer can be represented as 
where 
is any integer. The following table shows values of even number for different values of
where is any integer. The following table shows values of even number for different values of |
Even Number (
) |
0
|
0
|
1
|
2
|
2
|
4
|
3
|
6
|
Odd integers are those that are not divisible by 2 i.e. when an odd number is divided by 2, there is always a remainder that is equal to 1. All odd integers must end in 1, 3, 5, 7, or 9 i.e. their unit’s digit must 1, 3, 5, 7, or 9. An odd integer can be represented as 
where is any integer. The following table shows values of even number for different values of
where is any integer. The following table shows values of even number for different values of | |
Odd Number (
) |
0
|
1
|
1
|
3
|
2
|
5
|
3
|
7
|
There are certain rules involving operations such as addition, subtraction, multiplication or division. Remembering them may be helpful to solve certain types of problems:
Addition and Subtraction Rules for Odd and Even Integers
The sum of any number of even integers is always even.
The sum of even number of odd integers is even, the sum of odd number of odd integers is odd.
The difference of two even integers is always even. Even 0 is an even integer for that matter.
The difference of two odd integers is always even.
The following Table summarizes these rules for addition and subtraction:
Even
|
Odd
| |
Even
|
Even
|
Odd
|
Odd
|
Odd
|
Even
|
The following table summarizes these rules for multiplication
Even
|
Odd
| |
Even
|
Even
|
Even
|
Odd
|
Even
|
Odd
|
When dividing two integers, it is impossible to determine whether the result will be an integer or a fraction/decimal. Therefore there are no hard and fast rules for division of integers.
Let us look at some examples where the above concepts are used:
Example:
If 
and 
are odd numbers, which of the following can never be an even number?
and are odd numbers, which of the following can never be an even number?
A) 
B) 
C) 
D) 
E) 
Solution:
This problem has to be solved using the process of elimination. Read the question carefully, it wants you to identify the expression that can never be an even number. Therefore any expression that is reduced to an even number can be eliminated.
Answer choice A) is incorrect. Since and are odd numbers, we know that must be even since sum of two odd numbers is always even
and are odd numbers, we know that must be even since sum of two odd numbers is always even
Answer choice B) is incorrect. must be even since 
is odd as well as are odd number. Since sum of two odd numbers is always even, answer choice B) is incorrect.
must be even since is odd as well as are odd number. Since sum of two odd numbers is always even, answer choice B) is incorrect.
Answer choice C) is obviously incorrect since 
must be even.
must be even.
Answer choice D) is correct. Since is odd, must be odd. In addition, regardless of the value of ,
must be even. Since the sum of odd and even number must be odd, answer choice D) is correct
is odd, must be odd. In addition, regardless of the value of , must be even. Since the sum of odd and even number must be odd, answer choice D) is correct
Answer choice E) is also incorrect, since must be a multiple of 
and hence must be an even number.
must be a multiple of and hence must be an even number.
The correct answer is D.
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