1. Even Number An integer, represented by the general expression 2n for (i.e., an integer), is called an even number.
The set E = {…, -2n, …, -4, -2, 0, 2, 4, …, 2n, …} consists of even numbers, and each of its elements is an even number.
2. Odd Number
An integer, represented by the general expression 2n + 1 for , is called an odd number.
The set O = {…, -(2n + 1), …, -3, -1, 1, 3, …, (2n + 1), …} consists of odd numbers, and each of its elements is an odd number.
The difference of two odd numbers is even, their sum is even, and their product is odd.
For an odd number T,
T + T results in an even number.
T – T results in an even number.
The result of T × T is an odd number.
The sum, difference, and product of two even numbers are even.
For an even number E,
E + E results in an even number.
E – E results in an even number.
The result of E × E is an even number.
The sum and difference of an odd number and an even number are odd, and their product is an even number.
For an odd number T and an even number E,
T + E results in an odd number
E + T results in an odd number
T – E results in an odd number
E – T results in an odd number
The result of T × E is an even number
Note 1: In the set of integers, if the result of a multiplication is even, at least one of the numbers being multiplied is even.
Note 2: In the set of integers, if the result of a multiplication is odd, both numbers being multiplied are odd.
Note 3: All positive integral powers of even numbers are even. This is because Note 1 applies. Thus, for a positive integer n and an even number E, the result of T is always an even number.
Note 4: All natural number powers of odd numbers are odd. This is because Note 2 applies. Thus, for a natural number n and an odd number T, the result of T is always an odd number.
Note 5: It would be incorrect to generalize the division operation in the same way as multiplication as mentioned above.
Note 6: Odd numbers and even numbers are made up of integers. There is no number that is both odd and even. Zero (0) is an even number.