Here’s an interesting little tidbit:
For any prime p except 2 and 5, the decimal expansion of 1/p repeats with a period that divides p-1.
The period could be as large as p-1, but no larger. If it’s less than p-1, then it’s a divisor of p-1.
Here are a few examples.
1/3 = 0.33…
1/7 = 0.142857142857…
1/11= 0.0909…
1/13 = 0.076923076923…
1/17 = 0.05882352941176470588235294117647…
1/19 = 0.052631578947368421052631578947368421…
1/23 = 0.04347826086956521739130434782608695652173913…
Here’s a table summarizing the periods above.
|-------+--------| | prime | period | |-------+--------| | 3 | 1 | | 7 | 6 | | 11 | 2 | | 13 | 6 | | 17 | 16 | | 19 | 18 | | 23 | 22 | |-------+--------|
For a proof of the claims above, and more general results, see Periods of Fractions.