Integers $n$ such that the largest factor of $n$ is $n-6$

Question

A good integer is defined if its largest factor is $n-6$. How many integers satisfy this statement?

I know that $7$, $9$ and $12$ satisfy this statement and 12 should intuitively be the upper limit of $n$. From this I can deduce that $n-6$ should be a factor of $6$. This is more gut feeling than actual maths though. I want to know if there is a more elegant way of solving this question instead of brute force and maybe even a general solution to questions of similar caliber. At the very least a way to put this is equation form. I reckon there should be one since this question is from a math competition I went to and their questions almost always have elegant answers.

Answer 1

Hint If $(n-6)$ divides $n$, since $n-6$ also divides $n-6$, $n-6$ must divide $n-(n-6)$.

Answer 2

Method 1

Let $\frac{n}{n-6}=k$ for some positive integer $k$, then

$$ k=1+\frac{6}{n-6} \Leftrightarrow n=7,8,9,12.$$ Since $n-6$ is the greatest factor of $n$, therefore $n$ is either $9$ or $12$.

Method 2

Assume that $n-6$ divides $n$, then there exists an integer $k$ such that $$n=k(n-6) \Leftrightarrow n=6+\frac{6}{k-1} \Leftrightarrow k=2,3,4,7 \Leftrightarrow n=12, 9,8,7$$ Therefore $n=9$ and $12$ are the integers such that the largest factor of $n$ is $n-6$.