<s>[INST] Find all positive integers such that: $n^2-3|3n^3-5n^2+7$ 
Find all positive integers such that: $$n^2-3|3n^3-5n^2+7.$$

I did the following:
$$n^2-3|3n^3-5n^2+7,n^2-3\Rightarrow n^2-3|3n^3-5n^2+7,3n^3-9n$$
$$\Rightarrow n^2-3|5n^2-9n-7,5n^2-15\Rightarrow n^2-3|9n-8$$
Now we must have:  $9n-8\geq n^2-3$.We should determine the sign of  $n^2-9n+5$, but the equation: $n^2-9n+5=0$ does not have integer roots.How can we proceed?
 [/INST] Yes, $n^2-9n+5\leq0$, which gives $1\leq n\leq8$ and check it.
$n=2$ is valid.
</s>