Christmas Contest - Individual Round - Problem 12

Consider the decimal expansion of a fraction \dfrac{1}{k}, where k > 1 is a positive integer where 2 \nmid k and 5 \nmid k. Now, it’s decimal expansion will repeat after a certain number of digits, which we’ll call it’s period.

For example, \dfrac{1}{137} = 0.\overline{00729927}, so \dfrac{1}{137} has a period of 8.

Find the minimum value of k such that \dfrac{1}{k} has a period of 8.