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?

You can also say that 12 is even because there exist a number 6, such that 12 = 2 × 6. Having said that, we can come up with a formal definition.

This is the 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. 

How to quickly check if a number 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. For example, 98716 is an even number because its last digit shown in red is 6.

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

The figure below shows the difference between even numbers and odd numbers.

What are odd numbers?

You can also say that 27 is odd because there exist a number 13, such that 27 = 2 × 13 + 1. Having said that, we can come up with a formal definition.

This is the 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.

How to quickly check if a number 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

Basic operations with even and odd numbers

Adding two even numbers

even + even = even

4 + 2 = 6

Generally speaking, let n₁ = 2k₁ be an even number and let n₂ = 2k₂ be another even number.

n₁ + n₂ = 2k₁ + 2k₂

n₁ + n₂ = 2(k₁ + k₂)

Adding an even number to an odd number

even + odd = odd

6 + 3 = 9

Generally speaking, let n₁ = 2k₁ be an even number and let n₂ = 2k₂ + 1 be an odd number.

n₁ + n₂ = 2k₁ + 2k₂ + 1

n₁ + n₂ = 2(k₁ + k₂) + 1

Adding two odd numbers

odd + odd = even

13 + 13 = 26

Generally speaking, let n₁ = 2k₁ + 1 be an odd number and let n₂ = 2k₂ + 1 be another odd number.

n₁ + n₂ = 2k₁ + 1 + 2k₂ + 1

n₁ + n₂ = 2(k₁ + k₂) + 2

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

If you have any questions about this even and odd numbers lesson, just contact me.

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