Question on recurring decimal digits In my discrete maths class, I have come across an interesting

So, let me begin by writing some dull equations:
$\frac{1}{13}= 0.0769230769230$
$\frac{10}{13}= 0.769230769230$
$\frac{100}{13}= 7.69230769230$
$\frac{1000}{13}= 76.9230769230$
$\frac{10000}{13}= 769.230769230$
$\frac{100000}{13}= 7692.30769230$
Of course, a multiple of 10 only moves the sequence about. Imagine, though, that we take the integer component out of each of these equations. When they each start from a separate location and repeat the same pattern, they become instantly interesting:
$\frac{1}{13}- 0 = \frac{1}{13}= 0.0769230769230$
$\frac{10}{13}- 0 = \frac{10}{13}= 0.769230769230$
$\frac{100}{13}- 7 = \frac{9}{13}= 0.69230769230$
$\frac{1000}{13}- 76 = \frac{12}{13}= 0.9230769230$
$\frac{10000}{13}- 769 = \frac{3}{13}= 0.230769230$
$\frac{100000}{13}- 7692 = \frac{4}{13}= 0.30769230$
Hey, those equations are all very cool. They have the same characteristic of having the numbers appear in a shifted order, which isn't very clear if you just write out the fraction without understanding where it comes from.
Notice that this only works for the numerators $1 , 3 , 4 , 9 , 10 ,$ and $12$. The reason these numbers are special is that they are the powers of $10$ mod $13$. More generally, we could replace $13$ by any number $n$ coprime to $10$ and say that this is true of the powers of $10$ mod $n$. This is particularly interesting when every integer between $1$ and $n - 1$ is a power of $10$ mod $n$, meaning that every non-integer fraction would have this property (which is true of $n = 7$ and other numbers)