Proof of the divisibility rule of 17.

Rule: Subtract 5 times the last digit from the rest of the number, if the
result is divisible by 17 then the number is also divisible by 17.

How does this rule work? Please give the proof.

Answer

Write your number $10a+b$.

Then because 10 and 17 are relatively prime,
$$17\mid a-5b \iff 17\mid 10a-50b \iff 17\mid 10a+b$$ The last equivalence is because $10a+b-(10a-50b) = 51b$ is always a multiple of 17.