Factors of Repunits

Theorem
The decimal representations of the reciprocals of the first few prime numbers are as follows:


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! align="right" style = "padding: 2px 10px" | $R_n$ ! align="left" style = "padding: 2px 10px" | Factors of $n$ ! style = "padding: 2px 10px" | Also see
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 * align="left" style = "padding: 2px 10px" | $11$
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 * align="right" style = "padding: 2px 10px" | $4$
 * align="left" style = "padding: 2px 10px" | $11 \times 101$
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 * align="right" style = "padding: 2px 10px" | $5$
 * align="left" style = "padding: 2px 10px" | $41 \times 271$
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 * align="left" style = "padding: 2px 10px" | $3 \times 7 \times 11 \times 13 \times 37$
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 * align="right" style = "padding: 2px 10px" | $7$
 * align="left" style = "padding: 2px 10px" | $239 \times 4649$
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 * align="right" style = "padding: 2px 10px" | $8$
 * align="left" style = "padding: 2px 10px" | $11 \times 73 \times 101 \times 137$
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 * align="right" style = "padding: 2px 10px" | $9$
 * align="left" style = "padding: 2px 10px" | $3^2 \times 37 \times 333 \, 667$
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 * align="right" style = "padding: 2px 10px" | $10$
 * align="left" style = "padding: 2px 10px" | $11 \times 41 \times 271 \times 9091$
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 * align="right" style = "padding: 2px 10px" | $11$
 * align="left" style = "padding: 2px 10px" | $21 \, 649 \times 513 \, 239$
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 * align="right" style = "padding: 2px 10px" | $12$
 * align="left" style = "padding: 2px 10px" | $3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901$
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 * align="right" style = "padding: 2px 10px" | $13$
 * align="left" style = "padding: 2px 10px" | $53 \times 79 \times 265 a, 371 \, 653$
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 * align="right" style = "padding: 2px 10px" | $11$
 * align="left" style = "padding: 2px 10px" | $21 \, 649 \times 513 \, 239$
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 * align="right" style = "padding: 2px 10px" | $12$
 * align="left" style = "padding: 2px 10px" | $3 \times 7 \times 11 \times 13 \times 37 \times 101 \times 9901$
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 * align="right" style = "padding: 2px 10px" | $13$
 * align="left" style = "padding: 2px 10px" | $53 \times 79 \times 265 a, 371 \, 653$
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 * }
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 * }