Does this equation always spit out a natural number?

Question

I was messing around with WolframAlpha when I found that, apparently,
$$\frac{(10^n-1)}{s}$$ is a natural number for some natural number n and any odd s not divisible by 5. For example,
For s = 137, n = 8
For s = 29, n = 28
For s = 61, n = 60
For s = 49, n = 42
etc.
Curiously, should s be prime, $$n=s-1$$ It holds true for s = 3, 7, 11, 13, 17, 19, 23, 29, 61, 137, 2017, 9887.
I've taken a crack at proving/ disproving this to no avail. Could anyone here enlighten me?