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Determine for what values of $n$ the number $\frac{n+7}{2n+1}$ is an integer

Here's what I've tried

I think I solved the problem just for the positive integers:

Since $\frac{n+7}{2n+1}$ is a natural number (in this case) $$n+7\leq 2n+1$$ $$n\leq6 \rightarrow 0\leq n \leq 6$$

And the values that can only fit that satisfy that condition are $n=0$ and $n=6$ and those are the only values that generate a positive value for $\frac{n+7}{2n+1}$. But I have no idea how to find the negative values. How do I solve for the other cases? Is there another method to express all the solutions?

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Hint: $$\dfrac{n+7}{2n+1} = \dfrac{1}{2} + \dfrac{13}{4n+2}$$ So if $4n + 2 > 26$ or $4n + 2 < -26$, ...

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Since $$2(n+7)=(2n+1)+13,$$ $2n+1$ divides $n+7$ if and only if it divides $13$. Thus $2n+1$ is either $\pm1$ or $\pm13$.

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