Divisibility criteria of number 7 How do I prove the divisiblity criterium of number seven having $a_{n}*10^n + a_{n-1} *10^ {n-1} +  ... + a_3*10^3 + a_2*10^2 + a_1*10 + a_0$
For example, I underestand that in the divisibility criterium of 3 we have to use mod 3 to do it and the expression I wrote before is congruent (in mod 3) $a_{n} + a_{n-1} +  ... + a_3 + a_2 + a_1 + a_0$, but I don't find it for the number seven. I'd appreciate so much a bit of help.

Hint:
$10^3\equiv -1 \mod 7$ so you have to cut the number in slices of $3$ digits, starting from the right and check if the alternating sum of the $3$-digit numbers is divisible by $7$.
Another criterion:: 
Write the number $N$ as $10n+a_0,$ $\;0\le a_0\le 9$. $N$ is divisible by $7$ if $n-2a_0$ is.
