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?

Guest Oct 4, 2022

#2**0 **

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?

Guest Oct 4, 2022

#4**+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

Melody Oct 6, 2022