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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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