Definition 1

With the notation of Theorem 1, r is called the remainder of the division of a by b. If $r = 0$, we say that a is a multiple of b, or that a is divisible by b, or that b is a divisor of a, or that b divides a, or that b is a factor of a. The number q is then called the quotient of a by b and is denoted by $\dfrac{a}{b}$ or $a/b$.

If $a$ is not a multiple of $b$, the number $q$ is called the integral part of the quotient of a by b (cf. General Topology, Chapter IV, § 8, no. 2).

In this chapter, writing $a/b$ or $\dfrac{a}{b}$ will imply that $b$ divides $a$.

The relations $a = bq$ and $q = a/b$ are equivalent (if $b > 0$). Every multiple $a'$ of a multiple $a$ of $b$ is a multiple of $b$, and

Also, if $c$ and $d$ are multiples of $b$, then $c+d$ and $c-d$ (if $d\leq c$) are multiples of $b$, and we have

The integers which are multiples of 2 are said to be even, and the others odd. By Theorem 1 the odd integers are of the form $2n+1$.