Simple Proof Methods

The concept of divisibility is fundamental in number theory and discrete mathematics. We already, in fact, talked about divisibility when we mentioned even and odd integers which are divisible (or, not divisible) by $2$.

Definition. [Divisibility] An integer $m$ is divisible by an integer $n \neq 0$ if there exists a $k \in \mathbb{Z}$ such that $m = k \cdot n$.

We denote this by $n \mid m$ (

$n \mid m$

)), which reads as "$n$ divides $m$", "$m$ is divisible by $n$", or "$m$ is a multiple of $n$".

For example, $12$ is divisible by $3$ because there exists a $k \in \mathbb{Z}$, specifically $4 \in \mathbb{Z}$, such that $12 = 4 \cdot 3$, so we write $3 \mid 12$.

On the other hand, $10$ isn't divisible by $3$ because there exists no $k \in \mathbb{Z}$ for which $10 = 3k$. This is because $k = \frac{10}{3} = 3.\overline{3}$, which isn't an integer. As such, to show an indivisibility, we say $3 \not\mid 10$ (

$3 \not\mid 10$

).

[BTW, the bar over the digit $3$ after the decimal point in the number $3.\overline{3}$ (

$3.\overline{3}$

) means that $3$ keeps repeating infinitely after the decimal point: $3.33333\dots$.]