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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
avatar+76 
-5

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
avatar+114511 
+1

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 ..................   frown

 

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
avatar+76 
-5

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
avatar+114511 
+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
avatar+3 
+2

Thank you very much for taking the time to answer my question

 Jul 30, 2021

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