Form a valid conclusion, using all the statements for each argument. Give reasons.

If 5 divides a positive integer N and if N is greater than 5, then N is not a prime number.

N is a prime number.

 5 , _____ (divide/does not divide) a positive integer N or _____(N is composite number/ N is not greater than 5).

_______ (direct/indirect) statement.

We are given the argument:

1. **If 5 divides a positive integer \(N\)** and **\(N > 5\)**, then **\(N\) is not a prime number.**
2. **\(N\) is a prime number.**

Now, let's analyze the situation and form a conclusion:

### Step 1: Reword the first statement in logical form.
- The first statement says:  
  **If** 5 divides \(N\) **and** \(N > 5\), **then** \(N\) is not a prime number.  
  This can be written as:
  \[
  (5 \mid N) \land (N > 5) \implies N \text{ is not a prime number}.
  \]

### Step 2: Use the second statement.
- The second statement says that **\(N\) is a prime number**.  
  This means the conclusion of the first statement ("\(N\) is not a prime number") cannot be true.

### Step 3: Apply logical reasoning.
- If the conclusion of the first statement is false (because \(N\) is prime), then one of the conditions in the premise of the first statement must be false.
- The premise of the first statement involves two conditions:  
  (a) \(5 \mid N\) (5 divides \(N\))  
  (b) \(N > 5\)

### Step 4: Conclusion.
- Since \(N\) is prime and the first statement would lead to \(N\) not being prime if both conditions were true, **one of the conditions must be false**.
- The false condition could either be that **5 does not divide \(N\)** or **\(N \leq 5\)**.

### Final conclusion:
- **5 does not divide \(N\)** or **\(N \leq 5\)**.
- This is an **indirect statement** because we are negating one of the conditions based on the information that \(N\) is prime.

### Answer:
1. **5 does not divide a positive integer \(N\)** or **\(N \leq 5\)**.
2. This is an **indirect statement**.

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Do you need more details or have any questions?

Here are 5 related questions to expand the understanding:
1. What are the conditions for a number to be divisible by 5?
2. Can a number greater than 5 be prime if it is divisible by 5?
3. What are the properties of composite numbers?
4. How can indirect statements help in logical reasoning?
5. What is the difference between direct and indirect proofs in mathematics?

**Tip:** Prime numbers greater than 5 can never be divisible by 5 since a prime number only has two divisors: 1 and itself.

Form a valid conclusion, using all the statements for each argument. Give reasons.

If 5 divides a positive integer N and if N is greater than 5, then N is not a prime number.

N is a prime number.

5 , _____ (divide/does not divide) a positive integer N or _____(N is composite number/ N is not greater than 5).

_______ (direct/indirect) statement.

Explore a logical reasoning problem involving prime numbers and divisibility by 5. Learn how to analyze conditions, apply divisibility rules, and form valid conclusions. Suitable for high school students studying advanced algebra and number theory.

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