Properties of Divisibility
📌 Core Definition
An integer $a$ is divisible by an integer $b$ (where $b \neq 0$) if there exists an integer $k$ such that:
$a = b \cdot k$
We write this as $b \mid a$ (read: "$b$ divides $a$"). If $b$ does not divide $a$, we write $b \nmid a$.
🔑 The Four Fundamental Properties
Every non-zero number $a$ is divisible by itself.
Why? Because $a = a \cdot 1$, and $k = 1$ is an integer.
Examples:
$7 = 7 \times 1 \Rightarrow 7 \mid 7$ ✅
$-5 = (-5) \times 1 \Rightarrow -5 \mid -5$ ✅
Zero is divisible by any non-zero number $b$.
Why? Because $0 = b \cdot 0$, and $k = 0$ is an integer.
Examples:
$0 = 3 \times 0 \Rightarrow 3 \mid 0$ ✅
$0 = (-8) \times 0 \Rightarrow -8 \mid 0$ ✅
If $b \mid a$ and $c \mid b$, then $c \mid a$.
Why? If $a = b \cdot k_1$ and $b = c \cdot k_2$, then:
$a = (c \cdot k_2) \cdot k_1 = c \cdot (k_1 \cdot k_2)$
Since $k_1 \cdot k_2$ is an integer, $c \mid a$.
Example:
$3 \mid 6$ and $6 \mid 24 \Rightarrow 3 \mid 24$ ✅
If $b \mid a$ and $a \mid b$, then $a = b$ or $a = -b$.
Why? If $a = b \cdot k_1$ and $b = a \cdot k_2$, then:
$a = (a \cdot k_2) \cdot k_1 = a \cdot k_1 k_2$
So $k_1 k_2 = 1$, meaning $k_1 = k_2 = 1$ or $k_1 = k_2 = -1$.
Example:
$3 \mid (-3)$ and $(-3) \mid 3 \Rightarrow$ they are opposites ✅