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# number theory

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There exist several positive integers x such that 1/(x^2 + 2x) is a terminating decimal. What is the second smallest such integer?

Jul 28, 2021

#1
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For starters, only numbers that divide over 2 or 5 are terminating decimals.

It is not hard to see that 2 is much smaller than 5, so trying x=2 and x=4 and we get 4 is the second smallest such integer.

Hope this helps.

Jul 28, 2021
#2
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Hi Orange Juice,

"For starters, only numbers that divide over 2 or 5 are terminating decimals."

What about 4 or 8 or any power of 2 or any power of 5 or any power of 10 .................. Ohhh ....  do  you mean the denominator must be a power of 2 or a power of 5 or a power of 10 .........

Mmmm not sure what you mean ..............

Melody  Jul 28, 2021
edited by Melody  Jul 28, 2021
#4
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Just to clarify what I meant by that:

Fractions whose denominator does not include 5 and 2 as factors are non-terminating decimals unless the numerator can cancel the factor that is not 2 or 5.

And this chart would be great to check out too. Note that the four colors are different classifications of the examples.

OrangeJuicy  Jul 30, 2021
#5
+1

Mmm

so you are saying that the only prime factors of the denominator must be 2 and or 5.   .......

So    5^2*2^3        would be ok.

Melody  Jul 30, 2021
#3
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Thank you very much for taking the time to answer my question

Jul 30, 2021