Integer division
Given two integers a and b, we say that a divides b if there exists an integer c such that the product ac is equal to b. Equivalently, $$ \frac{b}{a} = c \rightarrow b = a \cdot c $$
When an integer a divides an integer b, this relationship is denoted by
$$ a \mid b $$
where
Note. If a is a divisor of b, then it also divides every multiple k of b, $$ a \mid k \cdot b $$ where k is any integer.
Common divisors
Two integers a and b may share a common divisor c belonging to the set of integers Z.
$$ c \mid a \\ c \mid b $$
Note. If c is a common divisor of a and b, then it also divides any integer that can be written in the following linear combination: $$ c \mid ( k \cdot a + j \cdot b ) $$ where k and j are integers.
Proof
If c divides a, then there exists an integer k such that ck = a. $$ ck = a $$ If c divides b, then there exists an integer j such that cj = b. $$ cj = b $$ For any integers s and t, $$ c \mid sa + tb \\ c \mid s(ck) + t(cj) \\ c \mid c(sk + tj) $$ It follows immediately that c divides c(sk + tj).
Units in the integers
If an integer a divides the number 1, then a is called a unit. $$ a \mid 1 $$
It is immediately apparent that, within the set of integers Z, only two integers divide 1.
These are a = 1 and a = -1.
$$ \frac{1}{a} \begin{cases} a=1 \rightarrow \frac{1}{1} \\ a=-1 \rightarrow \frac{1}{-1} \end{cases} $$
Euclidean division of integers
An integer is not necessarily a multiple of another integer.
In such situations, integer division yields both a quotient and a remainder.
If the remainder is a positive integer, then a is not a divisor of b.
Given two integers a and b, with b≠0, there exist unique integers q and r, called the quotient and the remainder respectively, such that $$ a = b \cdot q + r $$
This procedure is known as Euclidean division.
The remainder is an integer satisfying the inequality
$$ 0 \le r < |b| $$
If the remainder is zero, then b divides a.
A practical example
Consider the two integers a = 2 and b = 9.
The integer a is not a divisor of b.
Therefore, dividing 9 by 2 produces a nonzero remainder.
$$ b = a \cdot q + r \\ 9 = 2 \cdot q + r \\ 9 = 2 \cdot 4 + 1 $$
The quotient of the division is 4, with remainder 1.
The remainder is a positive integer between 0 and |9|.
And so on
