Divisibility rules/Rule for 17 proof

If the rule is $5n_0-k$ (the opposite of the suggested rule $k-5n_0$ , but that is proven in turn, and $k=n_110^0+n_210^1+n_310^2+...$ ) we know that :

$5n_0-k\equiv 5n_0+16k\equiv 5n_0+33k\equiv 5n_0+50k\equiv n_0+10k$ , which is our original number.

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Divisibility rules/Rule for 17 proof