# Why can any fraction whose denominator has only 2s and 5s in its prime factorization can be written as a terminating decimal?

Why can any fraction whose denominator has only 2s and 5s in its prime factorization can be written as a terminating decimal?

This is 7th grade math question.  It's not a homework problem.

Thank you!
John

Any number you can divide by 2 or divide by 5 terminates (does not repeat).

A denominator of 10^n has only 2s and 5s in its prime factorization.
phoffric

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If this is a 7th grade problem, I assume that naseeam wants a very simple answer as first provided. ... Thinkpads_User
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"Why can any fraction whose denominator has only 2s and 5s in its prime factorization can be written as a terminating decimal?"
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To be perfectly correct you must define "fraction" a bit narrower.
9/3 terminates
3 does not have only 2s and 5s in its prime factorization, so whether 9/3 terminates is irrelevant.
"3 does not have only 2s and 5s in its prime factorization, so whether 9/3 terminates is irrelevant. "
true
But it is relvant when considering the converse (I know, that was not asked)
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All covering the same ground over and over again here.

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"All covering the same ground over and over again here."   Not quite

you said
"Any number you can divide by 2 or divide by 5 terminates (does not repeat). . "

but that is not really an answer to the question asked (which was Why?

furthermore consider a number (you said ANY number so try 1/3)

(1/3)/2 = 0.16666666....

does not terminate.   so at best you have to expand your answer a bit.