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To analyze the statement **"The product of three consecutive positive integers is divisible by 6"**, let's determine whether it is true or false with proper justification.

### Step 1: General Form of Three Consecutive Positive Integers
Let the three consecutive positive integers be:
\[ n, n+1, n+2 \]
(where \( n \) is a positive integer).

### Step 2: Divisibility by 6
For a number to be divisible by 6, it must be divisible by both 2 and 3. We will now check if the product \( n \cdot (n+1) \cdot (n+2) \) always satisfies these conditions.

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#### **Condition 1: Divisibility by 2**
- Among any three consecutive integers, at least one of them is even because integers alternate between odd and even.
- This guarantees that \( n \cdot (n+1) \cdot (n+2) \) is divisible by 2.

---

#### **Condition 2: Divisibility by 3**
- Among any three consecutive integers, at least one of them is a multiple of 3 (since every third integer is divisible by 3).
- This guarantees that \( n \cdot (n+1) \cdot (n+2) \) is divisible by 3.

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### Step 3: Conclusion
Since the product of any three consecutive integers is always divisible by both 2 and 3, it must be divisible by 6. 

Thus, the statement **"The product of three consecutive positive integers is divisible by 6"** is **true**.

### Justification with an Example
Take three consecutive integers, such as \( 4, 5, 6 \):
\[
4 \cdot 5 \cdot 6 = 120
\]
\( 120 \) is divisible by 6.

Similarly, for \( 1, 2, 3 \):
\[
1 \cdot 2 \cdot 3 = 6
\]
\( 6 \) is divisible by 6.

This holds for all sets of three consecutive integers.

---

Would you like additional examples or a deeper explanation?  
Here are 5 related questions for further exploration:  

1. What is the product of four consecutive integers, and is it always divisible by any specific number?  
2. How can the concept of least common multiple (LCM) be applied to this problem?  
3. Why is divisibility by 2 and 3 sufficient to prove divisibility by 6?  
4. Can this rule extend to negative integers as well?  
5. Is there a mathematical proof for this statement using modular arithmetic?  

**Tip:** When analyzing divisibility, always check the prime factors of the divisor to ensure the conditions are met systematically.

The product of three consecutive positive integers is divisible by 6. Is this statement true or false? Justify your answer.

Explore the proof that the product of three consecutive positive integers is always divisible by 6. This solution covers divisibility rules for 2 and 3, ensuring a clear understanding of number theory concepts. Suitable for students in Grades 6-8.

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