 # Is this fraction non-terminating? I recently stumbled upon an observation: the fraction x Yesenia Obrien 2022-07-03 Answered
Is this fraction non-terminating?
I recently stumbled upon an observation: the fraction $\frac{x}{y}$ terminates if and only if $y$ only has prime factors $2$ and $5$
For example:
$\frac{1}{20}=\frac{1}{2\cdot 2\cdot 5}=0.05$
$\frac{1}{6}=\frac{1}{2\cdot 3}=0.1\overline{6}$
I think this is true because fractions are in the form:
$\frac{a}{10}+\frac{b}{100}+\frac{c}{1000}+\dots$
$\frac{a}{2\cdot 5}+\frac{b}{2\cdot 2\cdot 5\cdot 5}+\frac{c}{2\cdot 2\cdot 2\cdot 5\cdot 5\cdot 5}+\dots$
How can I rigorously prove this?\
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HINT: You basically have it. To say that you have a terminating decimal means that
$\frac{x}{y}=\frac{m}{{10}^{s}}$
for some integer $m$ and positive integer $s$. Since the only factors of ${10}^{s}$ are ..., the only possible factors of $q$ are ... . (Why? What are you tacitly assuming about $x$ and $y$?)