Montessori Maths - Is there a divisibility rule for 7?

While I was presenting the divisibility rules and how to arrive at them and the kids were marvelling at the simplicity of it and having fun arriving at the rules. We did the rule for 2,3,4,5,6 and then 8. One fourth grader couldn't understand why 7 has no rule. 

Let's derive it she said. And here's what she did

Every thousand to be divisible by 7, needs a 1 more.

Every hundred gives away a 2 to be divisible, ie 98 is divisible and 2 is left over for every hundred. So the hundreds digit*2

Every tens digit leaves a remainder 3. So it's tens digit*3

And the units are taken
1s(+)
10*3(+)
100*2(+)
1000*1(-). 

(Notation - + Has. - needs)

So every thousand needs a 1, then ten thousands need ten, 8 hundred thousand will need an 800 and so on. This makes it very difficult to test beyond the thousands digit, so the test fails, she concluded.

I was amazed and took her work to our dear colleague who despite her  professed math phobia, saw further patterns and pointed it out. Why do ten thousands need ten they may need only 3..Ah!!

On discussing this possibility with my 4th grader, she immediately saw the pattern and comes up with this..
1s(+)
10*3(+)
100*2(+)
1000*1(-). 

10,000*3(-) 

100,000*2(-)
1,000,000*1(-)
10,000,000*3(-)
(Notations: + Has. - needs)

And so on in the same pattern

So all the negatives add up and are subtracted from the positives. If the result is divisible by 7 the number is divisible. If not it's not divisible.

 Phew these kids blow me away! And now she wants to test her idea.. Stay tuned