Here’s an interesting little tidbit:

For any prime

pexcept 2 and 5, the decimal expansion of 1/prepeats with a period that dividesp-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.

There are a few more properties which I almost understand just enough to ask a question:

Whatâ€™s going on with integer multiples of reciprocals of primes?

Take the fraction portion of n/7 (base 10) for all integers n. They are all the same sequence of digits in the same order, just offset. The same is true for (many) other bases. N/7 is trivial in base 7, 14, et cetera. The same happens for other primes (even 2 and 3) you just may have to view them in larger bases to see it. Why?