When are \(\displaystyle{\frac{{{1}}}{{{n}!}}}\) repeating decimal with single

fun9vk 2022-03-15 Answered
When are 1n! repeating decimal with single digit repetend
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Answers (1)

pintorreeqwf
Answered 2022-03-16 Author has 3 answers
Step 1
A positive rational number has terminating decimal expansion iff it can be expressed as x=A10k for some A,kN
A rational number has a length-1 periodic decimal expansion iff it's not in the form above, but 10xx=9x=A10k.
Then, if x=1m we require that m=2a5b3c with a0, b0 and c1
That is, it must be a multiple of 3, and it can be a multiple of 2 and 5, with no other ' factor.
Then, if m=n!, we must have n{3,4,5,6}
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