Let's analyze the statement carefully:

"For integers $a, b, c, d$ with $a \neq 0$, $b \neq 0$, if $a \mid c$ and $b \mid d$, then $ab \mid cd$."

We want to determine which of the provided statements are useful for proving or disproving this statement.

Step-by-Step Breakdown:

1. $a \mid c$ means $c = k_1 \cdot a$ for some integer $k_1$.

   • This simply means $c$ is divisible by $a$.

2. $b \mid d$ means $d = m \cdot b$ for some integer $m$.

   • This indicates that $d$ is divisible by $b$.

3. Now consider the expression $ab \mid cd$:

   • We want to see if the product $ab$ divides the product $cd$. Using the fact that $c = k_1 \cdot a$ and $d = m \cdot b$, we can substitute into $cd$: $cd = (k_1 \cdot a) \cdot (m \cdot b) = k_1 m \cdot ab$
   • So, $cd = k_1 m \cdot ab$, which is clearly divisible by $ab$ because the factor $k_1 m$ is just some integer.

Thus, the statement appears true, meaning $ab \mid cd$ is indeed valid.

Evaluating the given options:

1. "If $b \mid d$, then $b = m \cdot d$ for some integer $m$."

   • This statement is false. If $b \mid d$, we should have $d = m \cdot b$, not the other way around. This would not be directly helpful in proving the statement.

2. "If $a \mid c$, then $c = k \cdot a$ for some integer $k$."

   • This is true and useful. It directly follows the definition of divisibility and is required in the proof to express $c$ in terms of $a$.

3. "If $b \mid d$, then $d = m \cdot b$ for some integer $m$."

   • This is true and useful. This follows the definition of divisibility and is necessary for expressing $d$ in terms of $b$.

4. "If $a \mid c$ and $b \mid d$, then $cd = (k_1 \cdot c) \cdot (k_2 \cdot d)$ for some integers $k_1$ and $k_2$."

   • This is not correct. The correct way to express $cd$ using divisibility is $cd = k_1 m \cdot ab$, not the product of $c$ and $d$ themselves.

5. "If $a \mid c$, then $a = k \cdot c$ for some integer $k$."

   • This is incorrect. If $a \mid c$, it should be $c = k \cdot a$, not the other way around. This would not be useful.

6. "The statement is false for $a = 2$ and $c = 3$."

   • This is incorrect. The statement holds true for $a = 2$ and $c = 3$, as long as the other divisibility conditions hold. Therefore, this statement is misleading and not useful for proving or disproving the original statement.

Conclusion:

The useful statements for proving the original statement are:

• "If $a \mid c$, then $c = k \cdot a$ for some integer $k$."
• "If $b \mid d$, then $d = m \cdot b$ for some integer $m$."

Would you like to explore further details or have any specific questions?

Here are some related questions:

1. How can divisibility properties be used to prove the general case for any integers $a, b, c, d$?
2. What are some counterexamples for divisibility theorems?
3. Can we prove this kind of divisibility for other algebraic structures beyond integers?
4. How does this apply to prime numbers?
5. What role does the greatest common divisor play in divisibility?

Tip: Always double-check the order of factors in divisibility. It's important to note that divisibility is typically defined with the divisor on the right side.