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.

One thought on “Reciprocals of primes

  1. There are a few more properties which I almost understand just enough to ask a question:
    What’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?

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