# Even and odd numbers

Even and odd numbers are straightforward concepts. I will start easy, but I will try to challenge the topic a little bit. ## What are even numbers?

An even number is any number that gives a remainder of zero when divided by 2. For example, 12 gives a remainder of 0 when divided by 2, so 12 is even.

We saw in divisibility rules that a number is divisible by 2 or gives a remainder of zero if its last digit is 0,2,4,6,or 8.

Therefore, any number whose last digit is 0, 2, 4, 6, or 8 is an even number.

Other examples of even numbers are 58, 44884, 998632, 98, 48, and 10000000.

Formal definition of an even number:

A number n is even if there exist a number k, such that n = 2k  where k is an integer.

This is formal way of saying that if n is divided by 2, we always get a quotient k with no remainder. Having no remainder means that n can in fact be divided by 2.

## What are odd numbers?

An odd number is any number that gives a remainder of 1 when divided by 2. For example, 25 gives a remainder of 1 when divided by 2, so 25 is odd.

Again, we saw in divisibility rules that a number is divisible by 2 or gives a remainder of zero if its last digit is 0,2,4,6,or 8.

Therefore, any number whose last digit is not 0, 2, 4, 6, or 8 is an odd number.

Other examples of odd numbers are 53, 881, 238637, 99, 45, and 100000023

Formal definition of an odd number:

A number n is odd if there exist a number k, such that n = 2k + 1  where k is an integer.

This is formal way of saying that if n is divided by 2, we always get a quotient k with a remainder of 1. Having a remainder of 1 means that n cannot in fact be divided by 2.

## Basic operations with even and odd numbers

even + even = even

4 + 2 = 6

even + odd = odd

6 + 3 = 9

odd + odd = even

13 + 13 = 26

Multiplication

even × even = even

2 × 6 = 12

even × odd = even

8 × 3 = 24

odd × odd = odd

3 × 5 = 15

Subtraction

even − even = even

8 − 4 = 4

even − odd = odd

6 − 3 = 3

odd − odd = even

13 − 3 = 10

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