# Even numbers

An even number is an integer that can be divided by 2 such that there is a remainder of 0. In other words, any integer that is a multiple of 2 is even. Generally, even numbers are numbers that have a 0, 2, 4, 6, or 8 in the ones place.

Examples

Test whether the following integers are even.

1. 10:

10 ÷ 2 = 5

There is no remainder, so 10 is even.

2. 38:

19 × 2 = 38

38 is a multiple of 2, so it is even.

3. 41:

41 ÷ 2 = 20 R1

There is a remainder of 1, so 41 is odd.

## 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

- even ± even = even
- even ± odd = odd
- odd ± odd = even

### Multiplication

- even × even = even
- even × odd = 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

## Is 0 even or odd?

0 is an even number based on the definition of even numbers. An even number is any integer that is evenly divisible by 2, where "evenly divisible" means that the result has no remainder. In other words, the result is an integer. Since 0 is an integer, and 0 ÷ 2 = 0 (or 0 × 2 = 0), 0 fits the definition of an even number.

0 also fits the rules discussed above for even numbers. For example, any even number added to or subtracted from another even number, is even.

0 + 2 = 2; 2 is even

4 - 0 = 4; 4 is even

Since both of the above result in even numbers, 0 must also be even.

Yet another way to show that 0 is even involves a definition of evenness much like the one we've already discussed. Given a set of objects, if the set of objects can be arranged such that there are two equally sized groups of objects, then the number of objects is even. For example, if we have 8 objects, we can arrange them into two equal groups of 4 objects. The same is true of 0 in that we can arrange 0 objects in two equal groups of 0 objects, making 0 even.