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 x y terminates if and only if y only has prime factors 2 and 5
For example:
1 20 = 1 2 2 5 = 0.05
1 6 = 1 2 3 = 0.1 6 ¯
I think this is true because fractions are in the form:
a 10 + b 100 + c 1000 +
a 2 5 + b 2 2 5 5 + c 2 2 2 5 5 5 +
How can I rigorously prove this?\
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Answers (1)

amanhantmk
Answered 2022-07-04 Author has 17 answers
HINT: You basically have it. To say that you have a terminating decimal means that
x y = 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?)
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