Please determine which of the following are true and which are false; use Definition 3.2 to explain your answers. a. 3|100 b. 3|99 c. -3|3 d. -5|-5 e. -2|-7 f. 0|4 g. 4|0 h. 0|0

To determine which statements are true and which are false, let's recall the definition of divisibility:

Definition 3.2: We say that an integer $a$ divides an integer $b$ (denoted $a | b$) if there exists an integer $k$ such that $b = ak$.

Now, we'll analyze each statement:

1. Analyze Each Statement

1. Statement a: $3 | 100$

   • Check if there exists an integer $k$ such that $100 = 3k$.
   • $k = \frac{100}{3} \approx 33.33$ (not an integer).
   • False.

2. Statement b: $3 | 99$

   • Check if there exists an integer $k$ such that $99 = 3k$.
   • $k = \frac{99}{3} = 33$ (is an integer).
   • True.

3. Statement c: $-3 | 3$

   • Check if there exists an integer $k$ such that $3 = -3k$.
   • $k = \frac{3}{-3} = -1$ (is an integer).
   • True.

4. Statement d: $-5 | -5$

   • Check if there exists an integer $k$ such that $-5 = -5k$.
   • $k = 1$ (is an integer).
   • True.

5. Statement e: $-2 | -7$

   • Check if there exists an integer $k$ such that $-7 = -2k$.
   • $k = \frac{-7}{-2} = 3.5$ (not an integer).
   • False.

6. Statement f: $0 | 4$

   • By definition, $0 | b$ is only true if $b = 0$. Here, $b = 4$.
   • False.

7. Statement g: $4 | 0$

   • Check if there exists an integer $k$ such that $0 = 4k$.
   • $k = 0$ (is an integer).
   • True.

8. Statement h: $0 | 0$

   • By definition, $0 | 0$ is usually considered true.
   • True.

Summary of Results

• a. False
• b. True
• c. True
• d. True
• e. False
• f. False
• g. True
• h. True

Final Answers

• a: False
• b: True
• c: True
• d: True
• e: False
• f: False
• g: True
• h: True