Divisibility rules/Rule 2 for 7 proof
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.
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.