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We know that $\dfrac{189}{685}=0.27\overline{59124087}$. What is the smallest positive integer $n$ such that $\dfrac{1}{n}$ is equal to a decimal number with an $8$-digit repeating block?

 Oct 4, 2022
 #1
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Sorry, don't understand your question with all those slashes and $ signs !

 Oct 4, 2022
 #2
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I will simplify if for you

 

What is the smallest positive integer n such that 1/n is equal to a decimal number with an 8-digit repeating block?

 Oct 4, 2022
 #3
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Thank you for translating it !

 

The smallest positive integer ==73

 

1 / 73 = 0.01369863  01369863  01369863  01369863.........and so on.

Guest Oct 4, 2022
 #5
avatar+118132 
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You have not simplified your question, you have just written it properly.  As you should have done int the first place.

Latex coding is greast if you render it properly.  Otherwise it is just coding garbage.

Melody  Oct 6, 2022
 #4
avatar+118132 
+3

 

I know that if x and the denominator are relatively prime then

x/9 will have  one number repeating

x/99 will have 2

x/999 will have 3 digits repeating

..

x/99999999  will have 8 digits repeating

 

Now

factor(99999999) = 3^2 *11 * 73 * 101 *137

 

So n is going to have to be a multiple of 1 or more of these factors

 

1/3 only has 1 repeating digit

1/9 only has 2 repeating digitsw

1/11 only has 2 repeating digits

1/33 only has 2 repeating digits

1/99 only has 2 repeating digits

1/73 = 0.01369863 with all 8 digits repeating.

 

So the smallest n is 73  

 Oct 6, 2022
edited by Melody  Oct 6, 2022

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