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:

ie.

Next note the following:

...

Next we append to 3 the zero so we can find the remainding term:

...

...

...

...

...

...

...

...

...

where 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