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The image presents a mathematical proof construction task, where the goal is to prove that for any integer \( n \), the expression \( n^2 - n \) is divisible by 7. The proof is constructed by arranging the given statements in the correct logical order.

Let me first transcribe the key parts of the image for clarity and then I’ll help construct the proof:

### Given Statements to Arrange:

1. Subtracting the terms of the equation by factoring \( n^2 - n = n(n-1) \).
2. \( n(n-1) \) is the product of two consecutive integers.
3. Both \( n \) and \( n-1 \) are integers.
4. By definition, at least one of the integers \( n \) or \( n-1 \) is divisible by 7.
5. Dividing both sides of this equation by 7 yields \( \frac{n^2 - n}{7} = \frac{n(n-1)}{7} \).
6. The definition of divisibility requires that \( n^2 - n \) is divisible by 7 if and only if \( n(n-1) \) is divisible by 7.
7. The sentence \( n^2 - n \) is divisible by 7 must be true.

### Correct Order to Prove the Statement:

1. \( n(n-1) \) is the product of two consecutive integers.
2. Both \( n \) and \( n-1 \) are integers.
3. By definition, at least one of the integers \( n \) or \( n-1 \) is divisible by 7.
4. Subtracting the terms of the equation by factoring \( n^2 - n = n(n-1) \).
5. Dividing both sides of this equation by 7 yields \( \frac{n^2 - n}{7} = \frac{n(n-1)}{7} \).
6. The definition of divisibility requires that \( n^2 - n \) is divisible by 7 if and only if \( n(n-1) \) is divisible by 7.
7. The sentence \( n^2 - n \) is divisible by 7 must be true.

This forms the logical progression to prove the statement.

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Would you like further explanations or details on any of these steps?  
Here are 5 related questions to further enhance understanding:

1. Why does the product of two consecutive integers guarantee divisibility by 7?
2. What is the general rule for factoring expressions like \( n^2 - n \)?
3. Can this method of proof be applied to other moduli besides 7?
4. How does the concept of divisibility tie into factoring in this context?
5. What are some other examples where such factorization helps in divisibility proofs?

**Tip**: When working with divisibility problems, factoring is often a useful approach, as it simplifies the structure of the expression into more manageable components.

Construct a proof for the statement 'For every integer n, the expression n^2 - n is divisible by 7' by arranging the given sentences in the correct order.

Learn how to construct a proof for the statement that n^2 - n is divisible by 7 for any integer n. This problem involves key number theory concepts, such as divisibility and properties of consecutive integers, making it suitable for students in Grades 9-11.

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