Number System : Properties of Even & Odd Numbers

by Wonder Boy · Published September 21, 2020 · Updated September 12, 2021

Let’s discuss about the result when basic arithmetic operations are carried out on Even & Odd Numbers.
1. Sum of two Even numbers is always an Even Number.
For Example:
4 + 6 = 10
12 + 8 = 20
6 + 18 = 24
(2n+2) + (2n+8) = 4n +10 which is an even number.

2. Sum of two Odd numbers is always an Even Number.
For Example:
3 + 5 = 8
15 + 17 = 32
13 + 3 = 16
(2n+1) + (2n+5) = 4n +6, which is an even number.

3. Sum of Odd number & Even Number is always an Odd Number.
For Example:
3 + 6 = 9 or 6 +3 = 9
5 + 10 = 15 or 10 + 5 = 15
9 + 12 = 21 or 12 + 9 = 21
(2n+1) + (2n +2) = 4n +3, which is an odd number.

4. Product of two Even Numbers is always an Even Number.
For Example:
4 x 6 = 24
2 x 8 = 16
56 x 4 = 224
$(2n+2)(2n+4) = 4n^2 + 12n + 8$ , which is an even number.

5. Product of two Odd Numbers is always an Odd Number.
For Example:
3 x 5 = 15
7 x 13 = 91
5 x 17 = 85
$(2n+1)(2n+3) = 4n^2 + 8n + 3$ , which is an odd number.

6. Product of Odd Number and Even Number is always an Even Number.
For Example:
3 x 6 = 18 or 6 x 3 = 18
5 x 8 = 40 or 8 x 5 = 40
7 x 8 = 56 or 8 x 7 = 56
$(2n+1)(2n+2) = 4n^2 + 6n + 2$ , which is an even number.

The above six properties may be summarized in the following table:

Addition	Example
Even + Even = Even	2 + 4 = 6
Odd + Odd = Even	3 + 7 = 10
Even + Odd = Odd + Even = Odd	2+7 = 9

Multiplication	Example
Even x Even = Even	2 x 4 = 8
Odd x Odd = Odd	3 x 7 = 21
Even x Odd = Odd x Even = Even	2 x 7 = 14

7. Power of an Even Number.
Since product of two even numbers is always an even number. Hence, power of any even number will result in an even number only.
$4^2 = 16$ , is an even number.
$4^3 = 64$ is an even number.
$a^n$ = a x a x a …. upto n times (where a is an even number) is also an even number.

8. Power of an Odd Number.
Since product of two odd numbers is always an odd number. Hence, power of any odd number will result in an odd number only.
$3^2 = 9$ is an odd number.
$3^3 = 27$ is an odd number.
$b^n$ = b x b x b …. upto n times (where b is an odd number) is also an odd number.

From above points 7 & 8, it can be easily concluded that the base of a number with power determines that the final result will be an odd number or even number.