Odd numbers
An odd number is an integer that cannot be evenly divided by 2. There is always a remainder when an odd number is divided by 2. If there is no remainder after division by 2, then the number is even. All integers are either even or odd. 0 is considered an even number. Odd numbers have 1, 3, 5, 7, or 9 in the ones place.
Examples
Test whether the following integers are even.
1. 11:
11 ÷ 2 = 5 R1
There is a remainder, so 11 is odd.
2. 87:
87 ÷ 2 = 43 R1
There is a remainder, so 87 is odd.
3. 184:
184 ÷ 2 = 92
There is a no remainder, so 184 is even.
Parity rules
Parity simply refers to the property of an integer being either even or odd. Any integer is either even or odd. Below are some properties of parity in the context of addition, subtraction, multiplication, and division.
Addition and subtraction
- odd ± even = odd
- odd ± odd = even
Multiplication
- odd × even = even
- odd × odd = odd
Division
Division is slightly different because the result of dividing whole numbers is frequently a fraction, rather than a whole number, meaning that it is neither even nor odd. Assuming that the quotient is an integer, it can only be even if the dividend has more factors of 2 than the divisor.
Example
Otherwise, any integer quotient will be odd.
There are an equal number of factors of 2, so the quotient cannot be even.
If we add another factor of 2 however:
The quotient becomes even.