How many natural solution this equation has?
$n^2 - 5n + \frac{6}{n} + 1$
My attempt is simple, the only "problem" this equation has is $\frac{6}{n}$ so the answer is $4$, $\{1,2,3,6\}$ but how can i guarantee that isn't there an another? For example for largers $n$? Maybe $n^2 -5n + 1$ completes $\frac{6}{n}$ and we have a natural number. I wasn't able to do it.. I tried to match the equation to a $k \in \mathbb{N} $, but i fails to get a another solution or proof that isn't.
Thank you
If $n$ is a natural number, then $n^2-5n+1$ is an integer, suppose $n\not \mid6$, then $\frac{6}{n} \notin \mathbb{Z}$ and hence $n^2-5n+\frac6n+1 \notin \mathbb{Z}$ and hence it can't be equal to $0$.
Hence, you have found all the natural solution.