Question 139
A positive integer $N$ is called a *good number* if the decimal representation of $N$ can be divided into at least 5 segments of digits, each segment containing at least one non-zero digit, and these segments (ignoring any leading zeros) can be viewed as positive integers which can be divided into two groups, with each group forming a geometric sequence in the appropriate order. (If a group has only one or two positive integers, it is also considered a geometric sequence.)
For example, 20240327 is a good number. In fact, it can be divided into $2|02|403|2|7$, a total of 5 segments of digits, with the two groups of positive integers (2, 2, 2) and (7, 403) each forming a geometric sequence.
Let $p = 1 + a + a^2 + \dots + a^m$ be a prime number, where $a \gt 1$ and $m \gt 2$ are integers. Prove that $\frac{10^{p-1}-1}{p}$ is a good number.
For example, 20240327 is a good number. In fact, it can be divided into $2|02|403|2|7$, a total of 5 segments of digits, with the two groups of positive integers (2, 2, 2) and (7, 403) each forming a geometric sequence.
Let $p = 1 + a + a^2 + \dots + a^m$ be a prime number, where $a \gt 1$ and $m \gt 2$ are integers. Prove that $\frac{10^{p-1}-1}{p}$ is a good number.