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?