a|b
𝑎
|
𝑏
  implies
Question 1Answer

a.
b=a.k
𝑏
=
𝑎
.
𝑘
 for some k∈Z
𝑘
∈
𝑍

b.
a=b.k
𝑎
=
𝑏
.
𝑘
 for some k∈Z
𝑘
∈
𝑍

c.
the quotient obtained when a
𝑎
 is divided by b
𝑏

d.
the remainder obtained when a
𝑎
 is divided by b

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 \( a = 0 \).

Explore the concept of divisibility in number theory with a focus on understanding what the notation a|b signifies. Learn how a divides b, and the implications of this mathematical relationship. Suitable for students in Grades 7-9.

Math Problem Statement

a. b=a.k 𝑏

b. a=b.k 𝑎

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