In mathematics, a **even number** is an integer that we can write in the form 2n (that is, it is exactly divisible by 2), where n is an integer. Conversely, integers that are not even are called **odd numbers** and we can write them as 2n + 1.

**Even number**

It is any integer that we can write in the form 2n, with n ϵ Z, or that we can divide exactly by 2. For example, the numbers 10254, -6,0,10, -4 are all even.

**Odd number**

It is any number that is not even, that is, it is an integer that we can write in the form 2n + 1, with n ϵ Z. For example, the numbers 10253, -5,1,9,15 are all odd.

**Properties**

**Addition and subtraction**

When adding (or subtracting) odd and even numbers the result is always:

Operation | Example |

Pair ± Pair = Pair | 6 + 4 = 10 -6 + 40 = 4 |

Pair ± Odd = Odd | 6 + 3 = 9 6 – 5 = 1 |

Odd ± Pair = Odd | 1 + 4 = 5 7 – 20 = –13 |

Odd ± Odd = Pair | 5 + 25 = 30 -7 + 15 = 8 |

**Multiplication**

When multiplying odd and even numbers the result is always:

Operation | Example |

Pair × Pair = Pair | 6 × 4 = 24 |

Pair × Odd = Pair | -6 × 5 = -30 |

Odd × Pair = Pair | 7 × (-20) = -140 |

Odd × Odd = Odd | -7 × 15 = 105 |

1. Determine which of the following numbers are even, expressing them in the form 2n, and which are odd, expressing them in the form 2n + 1, where n is an integer:

a) 3028

b) 44

c) 125

a) 3028 is an even number, since 2 × 1514 = 3028 where n = 1514. We can also determine that it is an even number because we can divide it exactly by 2.

b) 44 is an even number, since 2 × 22 = 44 where n = 22. We can also determine that it is an even number because we can divide it exactly by 2.

c) 125 is an odd number, since 2 × 62 + 1 = 125 where n = 62. We can also determine that it is an odd number because we cannot divide it exactly by 2 (it is not an even number).

2. Determine if the following operations result in an even number, or an odd number:

a) 48 ÷ 8

b) 56 × 23

c) 4 + 11

a) 48 ÷ 8 results in an even number, since:

48 / 8 = 6

What can we express as:

2 × 3 = 6 where n = 3

b) 56 × 23 results in an even number, since:

Even × Odd = Even

56 × 23 = 1288

What can we express as:

2 × 644 = 1288 where n = 644

c) 4 + 11 results in an odd number, since:

Even + Odd = Odd

4 + 11 = 15

What can we express as:

2 × 7 + 1 = 15 where n = 7