1. ## Repeating Decimals 1/7

Hello,

How would I go about explaining why all the fractions of seven consist of the same 6 digits cyclically permuted, and how would I explain why the first three digits and last three digits of the fraction add to 999? I believe it has something to do with Fermat's Little Theorem, but I'm not sure exactly how to explain these phenomenon. Any help is appreciated!

2. Hi,
in a fast attemp of solving the problem I noticed the following fact (that might be a rough proof).

First of all, division (when decimals exist) is based in the last digit a [0...9] with 7.
if it less than 7 must be done: $\displaystyle a0$
ie. $\displaystyle [10, 20, 30, 40, 50, 60]$

Next note the following:

$\displaystyle 10 mod 7=3$ ... $\displaystyle 10 floor 7=1$
Next we append to 3 the zero so we can find the remainding term:
$\displaystyle 30 mod 7=2$ ... $\displaystyle 30 floor 7=4$
...
$\displaystyle 20 mod 7=6$ ... $\displaystyle 20 floor 7=2$
...
$\displaystyle 60 mod 7=4$ ... $\displaystyle 60 floor 7=8$
...
$\displaystyle 40 mod 7=5$ ... $\displaystyle 40 floor 7=5$
...
$\displaystyle 50 mod 7=1$ ... $\displaystyle 50 floor 7=7$

where $\displaystyle floor$ is the integer part of the division.

There is a period at modulos of 7 with 10, 20, 30, 40, 50, 60

if you intefere more with that fact you can make a more elegant proof

3. Note that this can also be done via Primitive Root tecniques.