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The image contains a set of mathematical exercises focused on proving divisibility properties of sums of consecutive integers. Here's a summary of the problems:

1. **Prove that the sum of three consecutive integers is divisible by 3.**
   - **Hint**: Use the smallest of these integers to express the sum.

2. **Similarly, prove that the sum of five consecutive integers is divisible by 5.**

3. **Prove that the sum of seven consecutive integers is divisible by 7.**

4. **In general, prove that the sum of \( k \) consecutive integers is divisible by \( k \) if and only if \( k \) is odd.**
   - (a) Reduce the sum of \( k \) consecutive integers to a function of \( n \), where \( n \) is the smallest integer.
   - (b) Show that this sum is divisible by \( k \) only if \( k \) is odd.

5. **Prove that the sum of six consecutive integers is never divisible by 6.**

6. **Study the sum of \( 7 \) consecutive integers, noting that the sum is divisible by 7, and express it in terms of the smallest integer in the sequence.**

These exercises focus on arithmetic properties of consecutive numbers and divisibility, emphasizing modular arithmetic and algebraic manipulation.

Do you need help with any specific problem or all of them?

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Here are 5 relative questions to expand the information:
1. How can you generalize the result for \( n \) consecutive integers?
2. What role does modular arithmetic play in these proofs?
3. Can you prove similar divisibility properties for sums of consecutive even numbers?
4. How does the sum of \( k \) consecutive integers behave when \( k \) is even?
5. Can you extend these results to non-consecutive integers?

**Tip**: When working with sums of consecutive integers, expressing the integers in terms of a variable \( n \), such as \( n, n+1, n+2, \ldots \), can help identify patterns and simplify proofs.

Prove divisibility properties for sums of consecutive integers.

Learn how to prove divisibility properties of sums of consecutive integers. This problem covers key number theory concepts, such as divisibility and modular arithmetic, with examples for sums of 3, 5, 7 consecutive numbers.

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