# 41 is Smallest Number whose Period of Reciprocal is 5

Jump to navigation
Jump to search

## Theorem

$41$ is the first positive integer the decimal expansion of whose reciprocal has a period of $5$:

- $\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$

This sequence is A021045 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).

## Proof

From Reciprocal of $41$:

- $\dfrac 1 {41} = 0 \cdotp \dot 0243 \dot 9$

Counting the digits, it is seen that this has a period of recurrence of $5$.

It remains to be shown that $41$ is the smallest positive integer which has this property.

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |