139. Moving Digits. A certain number, which is written using fewer than 30 digits, starts (if we move from left to right) with the digits 15, that is, it has the form $15 \ldots$. If we multiply it by 5, the result can be obtained simply by moving these two digits to the right end; in the end, we get a number of the form ... 15. Find the original number.
139. Let $f$ be a proper fraction, in the decimal periodic expansion of which one period coincides exactly with the initial number $15 \ldots$ By the conditions of this problem, $5 f=$ $0, \ldots 15 \ldots 15 \ldots$, and $100 f=15, \ldots 15 \ldots 15$, from which $95 f=15$ and $f=\frac{3}{19}$. Expanding $\frac{3}{19}$ into a decimal fraction, we find that its period consists of 18 digits: 157894736842105263. This is the number we are looking for. It is unique if we consider numbers not exceeding 30 digits. If, however, we change this restriction and consider, for example, numbers that are written using no more than 50 digits, then there will be another solution to the problem - a 36-digit number, coinciding with two periods of the fraction $f$.

[H. T. R. Aude, A. M. M., 41, 268 (April 1934).]