generator of ideal in ring of fraction

Question

Z[1/17] is subring of Q consisting all fractions whose denominator is a power of 17. Every ideal of Z[1/17] is generate by an element of the form n/1, n belong to Z. Find the generator of ideal <4/17^8> as a smallest positive n such that n/1 generate the ideal. why the generator of the ideal can be expressed as n/1? I don’t really know how to approach the problem

Answer 1

It's just $n=4$, because the ideals $(4)$ and $\big(\frac4{17^8}\big)$ contain each other, since $\frac1{17^8}\in\Bbb Z[\frac1{17}]$.