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?
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?
+118609
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
