We want to prove the conjecture that the sum of any $n$ consecutive positive integers is a multiple of $n$. Specifically, we are to find the conditions under which the sum of $n$ consecutive positive integers is divisible by $n$.

Step 1: General Form of the Sum

Let the first number in the sequence be $x$. The next $n-1$ consecutive numbers will be $x+1, x+2, \ldots, x+(n-1)$. The sum of these $n$ consecutive integers is:

$S = x + (x+1) + (x+2) + \ldots + (x+n-1)$

This is an arithmetic series, where the first term is $x$ and the common difference is 1. The sum of an arithmetic series can be written as:

$S = \frac{n}{2} \cdot \left( \text{first term} + \text{last term} \right)$

In this case, the first term is $x$, and the last term is $x+n-1$. Therefore, the sum is:

$S = \frac{n}{2} \cdot \left( x + (x+n-1) \right)$

Simplifying this expression:

$S = \frac{n}{2} \cdot (2x + n - 1)$

Step 2: Divisibility by $n$

For $S$ to be divisible by $n$, the expression $\frac{n}{2} \cdot (2x + n - 1)$ must be divisible by $n$.

Notice that the term $n/2$ is a factor of $S$, so for divisibility by $n$ to hold, we need:

$2x + n - 1 \text{ to be divisible by } 2$

This means that $2x + n - 1$ must be an even number for $S$ to be divisible by $n$. This condition depends on the value of $n$.

Step 3: Concluding the Condition

We can conclude that the sum of $n$ consecutive integers will be divisible by $n$ if and only if $2x + n - 1$ is even.

Thus, the condition is that $2x + n - 1$ must be even, or equivalently:

$x \equiv \text{even number if } n \text{ is odd}$ $x \equiv \text{odd number if } n \text{ is even}$

Proof Summary:

The sum of $n$ consecutive integers starting at $x$ is given by:

$S = \frac{n}{2} \cdot (2x + n - 1)$

For $S$ to be divisible by $n$, we require $2x + n - 1$ to be even. This gives the condition:

• If $n$ is odd, then $x$ must be even.
• If $n$ is even, then $x$ must be odd.

This condition ensures that the sum is divisible by $n$.

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Would you like a more detailed breakdown of any of these steps? Feel free to ask any questions.

Here are some related questions you might find interesting:

1. How does the sum of consecutive integers relate to the sum of squares of integers?
2. What happens if we sum $n$ consecutive integers starting from a negative number?
3. Can this conjecture be generalized for any arithmetic sequence, not just consecutive integers?
4. How does the concept of modular arithmetic play a role in divisibility problems like this one?
5. What are some other number-theoretic properties that arise from summing consecutive integers?

Tip: When working with divisibility, it often helps to rewrite the sum in terms of its first and last terms. This can simplify your divisibility checks!