To start off, my apologies if this question may come off as illogical.
Let us take a natural number X and integral number Y and compute Y / X. We see that for values of X, such as 11 or 37, no matter what Y is, X will be a repeating number. Yes, I do realize that there are some values of Y that do not agree with this case, such as 3/12 (in which X = 12 and Y = 3). However, for the purpose of this question, we will consider 0.25 (the quotient of 3/12) as a repeating decimal (0.2500...). This is only for the purpose of the question.
Examples for 11 include 5/11 (0.4545...) and 22/11 (2.000...). 37 works as well. Examples include 9/37 (0.243...) and 11/37 (0.297). Now, moving onto the question:
What are the characteristics of numbers like X, beyond primes? Numbers such as 1, 2 (prime), 3 (prime), 5 (prime), 6, 8, 9, 10, 11 (prime), 12... even 37 (where any integral number Y produces a "repeating" decimal) but not 7, 13, 17, 19, 21, 23, 29...
Again, my apologies if this question seems badly worded/illogical. Curiosity formed this inquiry.
Answer
If you take $\frac{x_1...x_n}{9...9}$ where there are $n$ copies of $9$ in the denominator you'll get the repeating decimal $.x_1x_2...x_nx_1x_2..x_n...$ where the sequence $x_1x_2...x_n$ keeps repeating.
For example, $\frac{1234}{9999}=.12341234...$.
The numbers you point out all divide a number that consists of some sequence of $9$s.
$37\cdot27=999$ so $\frac{9}{37}=\frac{243}{999}$ and $\frac{5}{11}=\frac{45}{99}$, etc.
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