# Don't dare to try it!

Suppose we write the infinite decimal expansion for $$\dfrac1n$$ for any natural number $$n> 1$$ such that it is non-terminating. For example $$\dfrac12$$ can be expressed as $$0.4\overline9$$ as its infinite decimal expansion.

Denote $$v_p (n)$$ as the highest power of $$p$$ that divides $$n$$. Determine the length of the non-periodic part of the infinite decimal expansion of $$\dfrac1n$$.

Details and Assumptions

• As an explicit example, $$200 = 2^3 \times 5^2$$, so $$v_2 (200) = 3$$ and $$v_5(200) = 2$$ because $$2^3 | 200 , 2^4 \not | \ 200$$ and $$5^2 | 200, 5^3 \not | \ 200$$.
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