When does $n-2$ divide $n-5$?

Question

We got asked this question today in a national exam:

$n$ is a positive integer; find the values of $n$ for which $n-2$ divides $n-5$.

Answer 1

Hint: $$n-5=n-2-3$$ so we get $$\frac{n-5}{n-2}=1-\frac{3}{n-2}$$

Answer 2

$$\frac{n-5}{n-2}=1-\frac{3}{n-2}.$$ So we want the quantity $\frac{3}{n-2}$ to be an integer. Now ask yourself for what values of $n \in \{1,2,3,4, \ldots \}$ will $n-2$ divide $3$?