Let's address each of the statements one by one:

---

### (a) If \( a \mid b \) and \( c \mid d \), then \( ab \mid cd \).
#### **Analysis**:
- \( a \mid b \) means \( b = ka \) for some integer \( k \).
- \( c \mid d \) means \( d = lc \) for some integer \( l \).

Thus, we know:
- \( b = ka \)
- \( d = lc \)

Now consider the product \( cd \):
\[
cd = (ka)(lc) = kl(ac)
\]
This shows that \( cd \) is a multiple of \( ab \), because \( cd = (kl)(ab) \).

Hence, \( ab \mid cd \).

#### **Conclusion**:
The statement is **true**.

---

### (b) If \( a \mid bc \), then \( a \mid b \) or \( a \mid c \).
#### **Analysis**:
This statement suggests that if \( a \) divides the product \( bc \), then \( a \) must divide either \( b \) or \( c \).

This is **false**, and a **counterexample** will demonstrate this.

#### **Counterexample**:
Let \( a = 6 \), \( b = 2 \), and \( c = 3 \).

- \( bc = 2 \times 3 = 6 \).
- Clearly, \( a \mid bc \), because \( 6 \mid 6 \).

However:
- \( a = 6 \) does **not** divide \( b = 2 \) (since \( 6 \) does not divide \( 2 \)).
- \( a = 6 \) does **not** divide \( c = 3 \) (since \( 6 \) does not divide \( 3 \)).

Thus, \( a \mid bc \), but neither \( a \mid b \) nor \( a \mid c \).

#### **Conclusion**:
The statement is **false**, as shown by the counterexample.

---

### (c) If \( a \) and \( b \) are positive and \( a^2 \mid b^2 \), then \( a \mid b \).
#### **Analysis**:
The statement suggests that if \( a^2 \) divides \( b^2 \), then \( a \) must divide \( b \).

Let’s analyze this. If \( a^2 \mid b^2 \), then there exists an integer \( k \) such that:
\[
b^2 = k \cdot a^2
\]
This implies:
\[
b = a \cdot \sqrt{k}
\]
For \( b \) to be an integer, \( k \) must be a perfect square. Let \( k = m^2 \), so:
\[
b = a \cdot m
\]
Thus, \( a \mid b \).

#### **Conclusion**:
The statement is **true**.

---

### Final Summary:
- (a) True. Proven.
- (b) False. Counterexample provided.
- (c) True. Proven.

---

Do you want more details or have any questions? Here are some related questions:

1. Can you provide other counterexamples for statement (b)?
2. What happens if one of \( a \), \( b \), or \( c \) is negative?
3. How would the statements change if \( a \), \( b \), or \( c \) were not integers?
4. Can you provide examples where \( a^2 \mid b^2 \) but the quotient \( b/a \) is not 1?
5. What would happen if \( a = 0 \) in statement (a)?

**Tip**: When working with divisibility, always keep in mind how multiplication and factorization behave in relation to prime factorizations.

Explore proofs and counterexamples for divisibility properties involving integers. This detailed breakdown addresses three mathematical statements about divisibility and provides step-by-step reasoning. Suitable for advanced high school or undergraduate students.

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