Let’s formulate rules for addition, subtraction, multiplication, and division of Even and Odd numbers.
Here are the rules for adding / subtracting even and odd numbers:
Even ± Even = Even (e.g. 2 + 2 = 4; -4 – 2 = -6)
Odd ± Odd = Even (e.g. 1 + 1 = 2; -31 – 1 = -32)
Even ± Odd = Odd (e.g. 2 + 1 = 3; -12 – 1 = -13)
There are some very important observations here.
If you add or subtract any quantity of even numbers, the result will always be even.
For example,
-2 + 8 + 20 – 16 = 10 (even)
-2 + 8 + 20 – 16 + 8 – 18 – 2 = -2 (even)
This can be generalized as
Even ± Even ± Even ….± Even …. ± Even = Always Even
If you add or subtract an odd ‘quantity’ of odd numbers, the result will always be odd.
For example,
If you add or subtract five (odd ‘quantity’) odd numbers, like 3 + 11 – 19 + 7 + 9 = 11, the answer is odd
Similarly if you add or subtract three (odd ‘quantity’) odd numbers, like 3 + 5 – 17 = -9, the answer is odd
However, if you add or subtract an even ‘quantity’ of odd numbers, the result will always be even.
For example,
If you add or subtract six (even ‘quantity’) odd numbers, like 3 + 11 – 19 + 7 + 3 +1 = 6, the answer is even.
If you add or subtract four (even ‘quantity’) odd numbers, like 3 + 5 – 17 + 7 = -2, the answer is even.
This can be generalized as
Odd ± Odd ± Odd ….± Odd …. ± Odd = Odd or Even (depends on the ‘quantity’ of odd numbers being added or subtracted)
Here are the rules for multiplying even and odd numbers:
Even × Even = Even (e.g. 2 × 2 = 4)
Even × Odd = Even (e.g. -2 × 3 = -6)
Odd × Odd = Odd (e.g. 5 × 3 = 15)
Again, there are some very important observations.
Even times any other integer (odd or even) always comes out even. The only way for the product of numbers to be odd is if all of those numbers are odd.
For example,
-2 × 9 × 3 = -54 (even)
4 × 1 × 3 × 15 × 21 × 33 = 124,740 (even)
This can be generalized as
Even × Any integer × Any integer × Any integer…. × Any integer = Always Even
Odd × Odd × Odd × Odd …. × Odd ×…. Odd = Always Odd
Here are the rules for dividing even and odd numbers:
Even ÷ Even = Even or Odd or fraction (e.g. 100 ÷ 2 = 50, -14 ÷ 2 = -7, 2 ÷ 10 = 0.2)
Even ÷ Odd = Even or fraction (e.g. 10 ÷ 5 = 2, -20 ÷ 3 = -6.6)
Odd ÷ Odd = Odd or fraction (e.g. 15 ÷ 5 = 3, 9 ÷ 5 = 1.8)
Odd ÷ Even = fraction (e.g. 15 ÷ 2 = 7.5, 21 ÷ 4 = 5.25)
This brings us to the following rules regarding division of Even and Odd numbers:
When dividing Even and Odd numbers, there are multiple possibilities.
If there is one general rule that can be used for a specific case of division, it is this:
Odd/Even=Fraction