What's the smallest natural number that starts with $15$ that becomes $5$ times when these digits are moved to the end?
Obviously the original number must end in $3$ (so that $\times \, 5 = 15$). Also its third digit must be $7$ or $8$, since the third digit becomes the first digit of the altered number.
So if the intermediate part has a length of $n$ digits, the number must be of the form: $15XX\ldots X3$ where $XX\ldots X$ has $n$ digits.
Any ideas on how to continue?
Let all the $m$ digits after "15" form the number $y$. Prepending the "15" means $15\cdot10^m+y$, while appending "15" means $100y+15$, so we have the equation $$5\,(15\cdot10^m+y)=100y+15,$$ i.e. $$y=\frac{15\cdot10^m-3}{19}.$$ Now we can solve this with a bit of modular arithmetic, or like this:
$1/19=0.05263\color{red}{157894736842105263}1578947368421053\ldots$