Divisibility rules/Rule 2 for 7 proof
From AoPSWiki
Truncate the last digit of 
, double that digit, and subtract it from the rest of the number (or vice-versa). is divisible by 7 if and only if the result is divisible by 7.
Proof
An understanding of basic modular arithmetic is necessary for this proof.
The divisibility rule would be 
, where 
, where 
is the nth digit from the right (NOT the left) and we have 
and since 2 is relatively prime to 7, 
. Then yet again 
, and this is equivalent to our original number.
