Reciprocals of primes

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.