25. For what least $n$ in the decimal representation of the fraction $m / n$ can the digit sequence ...501... appear after the decimal point?
80.25. We can consider that $m / n=0.501 \ldots$ Let's examine the difference $m / n-1 / 2$. On one hand, $m / n-1 / 2 < 0.001$. On the other hand, $m / n-1 / 2 = (2m-n) / 2n$. Since $2m-n$ is an integer, the fraction $(2m-n) / 2n$ can only take values of the form $1 / 2n, 2 / 2n, 3 / 2n, \ldots$. Therefore, $1 / 2n < 0.001$, which implies $n > 500$. For $n=251$, the required fraction exists - it is $126 / 251$.