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Suppose we have a fraction $0 < \frac{a}{b} \leq 1$, where $a, b \in \mathbb{Q}, b \neq 0$. Can this fraction be written with another base in its denominator? Say I want to express this fraction with a denominator that its base is $3$. So the new denominator would be $3^n$ for some $n \in \mathbb{N}.$ Then can we find $c$ such that $\frac{a}{b} = \frac{c}{3^n}$ and $c \in \mathbb{N}$? For a concrete example, let $\frac{7}{10}$. How do we one find $n$ and $c$ such that $\frac{c}{3^n} = \frac{7}{10}?$

Josh
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  • For a concrete example, $c=\frac {21}{10}$ and $n=1$, and both the numerator and denominator $\frac {21}{10},3^1\in\mathbb Q$, $3^1\ne 0$. – peterwhy Sep 23 '22 at 02:33
  • That works, but I am interested in $c \in \mathbb{N}$. I will update the question. – Josh Sep 23 '22 at 02:40
  • In this example, $10c = 7\cdot 3^n$, the LHS is a multiple of $2$, but the RHS is not. – peterwhy Sep 23 '22 at 02:43
  • ok maybe it may not be possible for base $3$. What about other bases, such as $2$? I am interested to know what happens in general. Maybe the answer could be something like, "it is possible for some bases, but bot not for all". – Josh Sep 23 '22 at 02:46
  • Replacing the $3$ in $10c = 7\cdot 3^n$ by another integer base, then one requirement is that the base number should be a multiple of all the prime factors of $10$. – peterwhy Sep 23 '22 at 02:58

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