The notation $a \mid b$ (read as "a divides b") implies that $a$ is a divisor of $b$. This means that $b$ can be expressed as a product of $a$ and some integer $k$, i.e., $b = a \cdot k$ where $k \in \mathbb{Z}$ (the set of all integers).

Therefore, the correct answer is:

a. $b = a \cdot k$ for some $k \in \mathbb{Z}$

Explanation:

• $a \mid b$ means $a$ divides $b$, so there exists an integer $k$ such that $b = a \cdot k$.
• This implies that $b$ is a multiple of $a$, and the division $\frac{b}{a}$ yields an integer quotient.

Additional Questions:

1. What does it mean if $a$ does not divide $b$ (i.e., $a \nmid b$)?
2. Can the divisor $a$ be negative, and what does it imply if $k$ is negative?
3. If $a \mid b$, what can you say about the relationship between the greatest common divisor (GCD) of $a$ and $b$?
4. How is the divisibility property $a \mid b$ related to prime numbers?
5. If $a \mid b$ and $b \mid c$, what can be concluded about the relationship between $a$ and $c$?

Tip:

Remember that for any integer $a$, $a \mid 0$ because $0 = a \times 0$, but $0 \mid a$ is only true if $a = 0$.