**Natural numbers **The numbers 1, 2, 3, 4, 5, … are called the natural numbers. They are also called the positive integers.

**Cardinal and ordinal numbers **When natural numbers represent the quantity of something, they are called cardinal numbers. For instance, if we’re talking about 3 books, ‘3’ is a cardinal number. When they represent the order that something appears in, they are called ordinal numbers. For instance, if we’re talking about the 3rd book, ‘3rd’ is an ordinal number.

**Integers **The integers consist of the positive integers, the negative integers and 0.

**Divisibility **Let *a, b* be integers with *b* ≠ 0. If there exists an integer *q* such that *a = q · b*, we say that *b* divides *a* and write *b* | *a*. If no such *q* exists, we say that *b* does not divide *a* and write *b* ∤ *a*.

**Factors (divisors) and multiples** If an integer *a* is divisible by an integer *b* (*b* ≠ 0), we say that *a* is a multiple of *b* and *b* is a factor of *a*. Every integer is a multiple of ±1 and ±1 is a factor of every integer. Since 0 is a multiple of every integer, every non-zero integer is a factor of 0.

**Properties of the integers **Let *a, b, c, d*, … be integers

- If
*a*|*b*, then (-*a*) |*b*,*a*| (-*b*), (-*a*) | (-*b*), ∣*a*∣ | ∣*b*∣ - If
*a*|*b*and*b*|*c*, then*a*|*c* - If
*a*|*b*, then*a*|*bc* - If
*a*|*b*and*c*≠ 0, then*ac*|*bc* - If
*ac*|*bc*and*c*≠ 0, then*a*|*b* - If
*a*|*b*and*b*≠ 0, then ∣*a*∣ < ∣*b*| - If ∣
*a*∣ < ∣*b*∣ and ∣*b*∣ | ∣*a*∣, then*a*= 0 - If
*d*|*a*and*d*|*b*, then*d*| (*a*+*b*) and*d*| (*a*–*b)* - If , and among the (
*m*+*n*) terms (*m*+*n*– 1) of them are multiples of*d*, then the remaining term is also a multiple of*d*. - Among
*m*consecutive integers, at least one of them is divisible by*m*. - (
**Euclid’s division lemma**) For any two integers*a*,*b*with*b*≠ 0, there exist unique integers*q*,*r*such that*a*=*qb*+*r*, with 0 ≤*r*< ∣*b*∣.*a*|*b*precisely when*r*= 0

**Odd numbers **Integers which are not divisible by 2 are called odd numbers. Every odd number is of the form 2*k* + 1 for some integer *k*.

**Even numbers **Integers which are divisible by 2 are called even numbers. 0 is an even number. Every even number is of the form 2*k* for some integer *k*.