Recently, I have found this problem:
Given the fraction $\frac{5445469}{5445468}$, find the smallest base $b\;\in\;\mathbf{N}$ such that, in base $b$, the fraction has a finite number of decimal digits. To solve this problem, I have splitted the fraction into two terms as follows: $$\frac{5445469}{5445468}=\frac{5445468}{5445468}+\frac{1}{5445468}=1+\frac{1}{5445468}$$ Now, the first term is a $1$, so in every base it's always $1$ as the numerator of the fraction. We have to find the minimu base for which $\frac{1}{5445468}$ has a finite number of terms.
I know the factorization of $5445468$ that is: $$5445468=2^2\cdot3^4\cdot7^5$$ but how can we get $b$? I think that in order to have a finite decimal representation, $5445468$ in base $b$ has to be multiple of $2$ and $5$. Is it correct?
from Hot Weekly Questions - Mathematics Stack Exchange
Matteo
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